Incorporating the other polyhedral dice into crit table resolution.
My group already does it to save time. And, we move from exact number to 1st one on top and then next undamaged crit down style of counting. So, on an arm, for example, if you have 9 slots full, we would roll a d10, and reroll on a 10. (I'm now toying with the cycling idea.) And, as you take crits, you work your way down the dice sizes, as best fit. So, if after that 9-slot arm takes its first critical hit, the next two crits will be determined with a d8 because there are only 8 valid critical slots left. 7 crits left would still be resolved with a d8 with a potential reroll. Then a d6, and so on.
Although, a cyclical style of crit table resolution might work that doesn't require a reroll. To use the 9 slot arm example above, you would roll the two different colored d6s as normal, figuring out high-low first, and then if it was the lower box (4-6) and you rolled a 4, 5, or 6 which would hit a blank spot, you'd just cycle the count from the top of that section.
To save on Initiative rerolls: If you get a tie, check to see who has the highest individual d6 result. That breaks ties. If you're still matched, THEN you reroll.
Edit: As an example, you have two sides roll Init, and both roll a 7 on 2d6 as per standard. But, in this case, one roll is a combination of 3 and 4 while the other is a combination of 2 and 5. The result with the 5 wins.