guildsbounty and Tristian have capably discussed the heights of various races and what they might have to do to fit behind the wall. I will limit myself here to attempting to clarify a few rules interactions.
A few words about that wall
The object produced by the minor conjuration class feature can be up three feet on a side, so it should have no problem producing the wall mentioned by the querent. However, note in the description that it has a maximum weight of ten pounds, which is awfully light to serve as cover. At its maximum size of 27 cubic feet, it would be a cube of a mere 0.059 g / cc, or only slightly more dense than Styrofoam. This 'wall' could easily be knocked aside by any incoming attack, and most projectiles would bury themselves deep within it and start to break it apart. A wall still three feet long and high, but less deep, could be more dense, but could then be more easily knocked over or penetrated by a small diameter but high-energy attack.
Crouching is not Hiding1
At best, crouching behind the curiously light wall will make your elf Unseen and perhaps Heavily Obscured. However, neither of these is Hidden. Hidden is more than just Unseen (although you generally have to be Unseen to make an attempt to Hide). When you hide, you also attempt to remain unheard, unsmelled, undetected by any sense. Note that even invisible creatures are not (automatically) Hidden. Thus, you could be crouching behind the wall, Unseen, but other creatures still know that you are there - they can detect your presence by the noises you make (your breathing, the clinking of your armor, etc.)
Prone doesn't have to mean prostrate
guildsbounty opens their answer by stating "There are no rules for Crouching in the core ruleset. There is Standing and Prone, with nothing in between."
This is quite right, but since no other answers have yet pointed it out, it bears mentioning that not every prone PC is flat on the ground. It is true that the definition of prone states that it is "lying on the ground". However, prone is also a condition that can be imposed on, or perhaps even assumed by, creatures that are not on or even near the ground - for example, creatures that are underwater or in the air. Jeremy Crawford has said that
You can almost always be knocked prone. About the only time it's physically impossible for you to be knocked prone is when you're affixed to something that keeps you upright.
Thus the elf in question could be standing behind the wall, or it could be flat on the ground (prone in the strict sense) behind the wall, and RAW covers both cases. If the elf is in some other position though, such as crouching, squatting, sitting, kneeling, etc., then it falls on the DM to decide how to treat it. The effects of being prone are:
A prone creature's only movement option is to crawl...
The creature has disadvantage on attack rolls.
An attack roll against the creature has advantage if the attacker is within 5 feet of the creature. Otherwise, the attack roll has disadvantage.
A prone creature thus moves slower, has a more difficult time attacking, and is easier to attack when close, but harder to attack from a distance. All of these effects seem qualitatively true for someone who is crouching, squatting, sitting, kneeling, etc. Thus, the simplest thing to do is just to say - 'yes, you are crouching behind the wall and have total cover, but we will treat you as prone until you stand up.' If you can be prone in the air or underwater, then prone doesn't have to mean flat on the ground, and if prone doesn't have to mean prostrate we can apply the condition to other postures.
This is the approach that I use in my games, and it works fine. I have one player, in particular, whose characters drop prone at the first sign of missile fire, and on their turn stand, move (half), shoot, and drop prone again. Within the narrative they are not face down on the dungeon floor - rather they are crouching, kneeling behind low obstacles, hugging the wall, etc. I lump all of these defensive postures under the mechanic of being prone and the game plays fine.
In between standing and lying down
A DM who wants a more granular approach certainly wouldn't have to apply all the effects of being prone to someone who was in between standing and lying flat. They could apply some but not others, or apply them but to a lesser extent. Now we are in house rules territory. This stack discourages idea-generating questions and untried suggestions for answers. I will mention a few possibilities below, but they should be understood more as examples of how a DM might apply house rules to this situation rather than actual suggestions themselves. So long as the effects applied are somewhere from those of RAW standing, to RAW prone (inclusive) there are unlikely to be balance issues, so we have a nicely defined idea-space.
Crawling on all fours, or knee-walking (shikko), are faster than a belly-crawl but slower than walking - probably not fast enough to merit an increase in speed above crawling, but having it cost a half foot extra of movement for each foot traveled rather than a full extra foot would be a plausible intermediary.
Some missile weapons, such as shortbows and light crossbows, could reasonably be employed while kneeling without imposing disadvantage on attacks, while longbows could not. Piercing weapons like spears can be used effectively while kneeling, especially as part of a defensive formation. Even two-handed slashing weapons can be used effectively while kneeling - provided the wielder has trained specifically and extensively in the form. A DM house-ruling this could easily get into the weeds with which particular weapons and styles still incur full disadvantage, which might have half-disadvantage, and which are allowed at full effect. A simple compromise is to adapt an already-existing rule: The weapon properties tell us that "Creatures that are Small or Tiny have disadvantage on attack rolls with heavy weapons. A heavy weapon's size and bulk make it too large for a Small or Tiny creature to use effectively." We could simply say that Heavy weapons are used at disadvantage in any position other than standing, while weapons that are not Heavy can be used without penalty while kneeling.
Being prone when being attacked by ranged weapons presumably represents the difficulty in hitting a smaller profile. But then again, size does not directly affect AC. A kneeling human PC might be harder to hit then when they were standing, but then again they might still be taller than a standing gnome for whom there are not penalties. Thus a kneeling character attacked from beyond 5 feet could reasonably be awarded disadvantage (as if prone), half-disadvantage (as an intermediary), or no protection (as if standing). To the point of the original question, lowering your profile does seem like it would more effectively use cover, and the existing rules on cover (total, three-quarters, half) already require the DM's assessment of how much of the body is covered and how much revealed, so this decision will need to be made anyway.
While it is possible to defend oneself against melee attacks while kneeling, even the most dedicated practitioner of hanmi handachi has more limited mobility and reach than a standing defender. Giving the attacker at least half advantage if not the full advantage against a prone opponent seems reasonable.
1In fact, they made a whole movie about it. Spoiler: The tiger was crouching, while the dragon was hidden.