Rolls of 1 on all damage dice you roll are considered for Overkill, before the highest are kept
This is actually an intentional change from the prerelease version of the rules, which becomes more obvious if you compare the wording of the two.
The description of the Overkill tag in Prerelease Beta v2 (aka PR2) of the core rulebook reads (p. 131; emphasis mine):
When attacking with this weapon, if any of the final damage dice come up as ‘1’, take 1 heat, pick them up, and re-roll them. This process can repeat if you keep rolling ‘1’s.
However, this was changed in the official release of the core book to read as quoted in the question (p. 105; emphasis mine):
When rolling for damage with this weapon, any damage dice that land on a 1 cause the attacker to take 1 [Heat], and are then rerolled. Additional 1s continue to trigger this effect.
This is a clear change that affects how the Overkill tag is resolved; it refers to not only "final" damage dice, but "any" damage dice that show that result. This seems to indicate that all damage dice you roll are checked for 1s (triggering the Overkill weapon tag's effect and resolving it) before keeping only the highest results of the final dice rolls.
This especially matters for the IPS-N Vlad's Combat Drill
As noted in this issue report for PR2 about Overkill's interaction with critical hits and the discussion below it, this change does actually matter quite a bit for at least one weapon: the IPS-N Vlad's Combat Drill (p. 155), unlocked at license rank III of the Vlad.
The Combat Drill is a Superheavy Melee weapon with the Overkill tag that does 3d6 kinetic + 1d6 energy damage normally, and has this additional effect:
When attacking a character that is PRONE, IMMOBILIZED, or STUNNED, this weapon’s OVERKILL tag does an extra +1d6 bonus damage each time it activates. This can activate indefinitely if the new bonus die result is a 1, triggering OVERKILL again.
The apparent intent of Overkill is to effectively institute a minimum damage total (since you can't end up with any 1s), at the cost of taking heat. The Combat Drill adds an extra damage die on top of that, when the Overkill tag activates in the specified circumstances.
However, the PR2 version of the Overkill tag actually resulted on doing less damage on a crit than on a regular hit, according to an analysis by the user liq3, initially referenced by the poster of that issue report and then shared by liq3 themselves:
Someone else also noted that due to the way Overkill interacts with critical hits, it is harder for the Combat Drill to activate it's overpenetration on a critical hit, which seems counterintuitive. Whether this is a big enough deal to warrant a rules change is a developer decision.
I'm the one who ran numbers for it. The results always came out to something like this. The damage and heat are average, 1,000,000 damage rolls for each one.
Non-final Dice Crit - Damage: 29.174002 Heat:2.000419
Non-crit roll Damage: 19.99817 Heat:0.998748
Final dice crit Damage: 19.267168 Heat:0.007653
So, ironically, if you only explode the top 4 dice out of 8, it does less damage than a normal roll of 4 dice with exploding. Here's the python code I used to calculate that if anyone wants to check it. https://pastebin.com/he3j17pA
In essence, before accounting for Overkill, you'd roll 4d6 on a regular hit with this weapon, or 8d6 (keep highest 4) on a crit. Each time Overkill activates against a Prone, Immobilized, or Stunned target, you also add 1d6 bonus damage (in addition to adding 1 Heat and rerolling the damage die).
As can be seen from the analysis above, taking only the highest damage dice results first and then checking for results of 1 would cause Overkill to activate less often than checking for results of 1 and resolving Overkill first before keeping only the highest dice. As a result, in PR2, crits with the Combat Drill did less damage on average against qualifying targets than even regular hits, because the bonus damage had much less chance to apply.
The more damage dice you roll, the less likely there is to be a result of 1 in the damage dice kept on a crit. This effect is less noticeable for a weapon like the SSC Death's Head's Vulture DMR (which only does one die of damage and has no additional effect on Overkill), as seen in Medix2's answer - but it's very noticeable for a weapon like the IPS-N Vlad's Combat Drill. This may have partly motivated the change to Overkill between PR2 and the final release.