RAW says the DM resolves simultaneous events
Favored Foe says (emphases mine):
When you hit a creature with an attack roll, you can call on your mystical bond with nature to mark the target as your favored enemy for 1 minute or until you lose your concentration (as if you were concentrating on a spell).
The first time on each of your turns that you hit the favored enemy and deal damage to it, including when you mark it, you increase that damage by 1d4.
Your first use of Favored Foe takes your concentration, and this will immediately drop your concentration on hunter's mark, as per the spell concentration rules and the clarification of those rules in the SAC. However, your first use of Favored Foe starts "when you hit", so you have two other immediate consequences as well: your base damage (applied "on a hit") and your hunter's mark damage bonus (which is applied "whenever you hit"). Thus your one event, hitting, triggers three different consequential effects - and it is up to the DM to decide the resolution order of these simultaneous effects. As soon as either of the two damage effects resolves, you also trigger another immediate effect - the Favored Foe damage (which occurs the first time on a turn you "hit...and deal damage"
The rules on Making an Attack say:
[A]n attack has a simple structure.
- Choose a target...
- Determine modifiers...
- Resolve the attack. You make the attack roll. On a hit, you roll damage, unless the particular attack has rules that specify otherwise. Some attacks cause special effects in addition to or instead of damage.
Step three, resolving the attack, contains all of the guidance we are given in how to apply simultaneous consequences of a hit - that is, practically none. "Some attacks cause special effects in addition to...damage" is all we have to work with. In particular, the 5e rules are very much unlike a game such as M:tG, where simultaneous effects have a precisely defined priority that determines their order of resolution.
Thus, your DM can rule that the resolution order of the four effects after the hit is: mark your Favored Foe, do base damage, do Favored Foe bonus damage, do hunter's mark bonus damage, concentration drops on hunter's mark.
Your DM could also rule that the resolution order is: mark your Favored Foe, drop concentration on hunter's mark, do base damage, do Favored Foe bonus damage, no hunter's mark bonus damage is done because at the time the hunter's mark bonus damage (triggered by the hit) resolves, there is no longer a hunter's mark in effect.
ANSWER: These events can be resolved in any order the DM decides, with the only limitation being that the Favored Foe bonus damage cannot occur until after one of the other damages has resolved. In particular, there is no priority given to dropping concentration effects occurring before the other consequences of the hit, unless your DM decides to do so.
Your DM could also decide to use the XGtE optional rule, which allows the player whose turn it is to decide the order of resolution of simultaneous events. In this case, you as the player of the ranger can certainly arrange them so that you get the bonus damage from hunter's mark once before your concentration on it drops.
But does Favored Foe even require concentration?
The answer above assumed that Favored Foe requires concentration, but I think that is at least questionable, given the vague wording of the feature and the uncertain of terms like "as if". Below I conclude that we can't say for sure whether Favored Foe requires concentration, but if you value consistency with other features then it should require concentration.
RAW it is unclear
The spell concentration rules are pretty clear - you cannot concentrate on two spells at once, so as soon as you begin concentrating on one, you lose concentration on any other on which you were concentrating. So the question is, does Favored Foe, not being a spell, actually require concentration? That depends on what the phrase 'as if' is modifying.
When you hit a creature with an attack roll, you can call on your mystical bond with nature to mark the target as your favored enemy for 1 minute or until you lose your concentration (as if you were concentrating on a spell).
As written, it is not clear how to apply the "as if".
"As if" could be modifying "your concentration". That is, it could be saying: You are not actually concentrating on a spell - but this class feature requires concentration like a spell does, and thus anything that breaks concentration will result in the early end of this effect. Also, you are effectively concentrating for the purposes of other effects that interact with concentration. If this is the intended interpretation, then using Favored Foe will immediately end your concentration on hunter's mark (as is assumed in the discussion above about simultaneous effects).
On the other hand, "as if" could be specifically modifying "lose your concentration". That is, it could be saying: You are not concentrating on a spell, and you are not even concentrating on the class feature - not in any sense that the rules define as 'concentration'. However, this class feature can be ended before its full duration by any effect that would cause you to lose concentration on a spell, as if you had been doing so. In other words, you do not actually concentrate to use the feature, but things that would make you lose concentration will make you lose the feature, too. If this is the intended interpretation, then using Favored Foe will not end your concentration on hunter's mark. Casting hunter's mark while you had Favored Foe running, however, would cause you to lose Favored Foe - you can lose Favored Foe, but using Favored Foe does not make you lose anything else.
There is no way, RAW, to decide which interpretation is correct because the language of Favored Foe is not clear.
RAI it probably does require concentration
There are a number of other class features, as well as magic items, which have similar "as if" language around non-spell uses of concentration, so we can look there for intent.
Invoke Duplicity (Trickery Domain Cleric Feature) lasts "1 minute, or until you lose your concentration (as if you were concentrating on a spell)."
Draconic Presence (Draconic Bloodline Sorcerous Origin Feature) lasts "1 minute or until you lose your concentration (as if you were casting a concentration spell)..."
Minor Alchemy (Wizard School of Transmutation Feature) lasts "1 hour, or until you lose your concentration (as if you were concentrating on a spell)..."
Dark Delirium (Archfey Patron Warlock Feature) lasts "1 minute or until your concentration is broken (as if you are concentrating on a spell)."
While all of these features are examples of "as if" concentration without spellcasting, none of them can help decide this question because they use the same or similar language. They all beg the same question of whether you are actually concentrating in a positive sense, or are just saying that the feature can be ended in the same ways that concentration is ended.
Fortunately, there are other features that are more clear.
Visions of the Past (Cleric Knowledge Domain Feature) says "You can meditate in this way for a number of minutes equal to your Wisdom score and must maintain concentration during that time, as if you were casting a spell."
Here we see that not only can the feature be lost as if breaking concentration, but that uses it requires maintaining concentration as if casting our spell. This is our first clue that features like this are supposed to work with all aspects of concentration, not just the mechanics for losing it.
Even more informative is a ring of djinni summoning, which says the summoned djinn "remains as long as you concentrate (as if concentrating on a spell), to a maximum of 1 hour..."
Here we see a very clear statement - you actually are concentrating, just as if you were concentrating on a spell. This language is also used in the ring of shooting stars: "The spheres last as long as you concentrate (as if concentrating on a spell), up to 1 minute."
So, RAW, we know how Visions of the Past and the two rings work, and we know that using any of them would end hunter's mark. Can we infer that all the other features, including Favored Foe, do the same? We can if we value coherency. RAW, Favored Foe is vague and requires DM resolution. If you think that all non-spell concentration effects should work the same as each other, then by RAI Favored Foe acts in all aspects like you were concentrating on it. With the text of Favored Foe as written, your DM could rule that using it does not even break your concentration on hunter's mark, but if they like to be consistent in such rulings, it should break concentration because other similar features do.