As many have noted, counties are not the critical unit in New England. In fact counties have been dissolved as governing bodies in CT, RI, and parts of MA. Towns (and cities when so named) are the fundamental unit. For these states keeping towns intact is far more important than keeping counties (or their historical boundaries) intact.
So, the challenge here is to divide the New England states into CDs while keeping towns intact and minimizing the average population deviation. Now that I have a statistical model from the whole-county states, I'll be interested in comparing to these states with far more jurisdictions to manipulate.
As before the rule requires no district to have more than a 0.5% deviation from the ideal, and point contiguity is not allowed. Bonus points for making districts that are connected by roads internally, but we won't let that stand in the way of a great plan, such as the near-perfect split of ID.
Also, some have suggested that NJ could be on this list as well, so I'll include it (boroughs, cities townships, etc all count as towns). However, I won't extend to NY and PA since both have cities that exceed the population of one CD, and can't really fit this rule.
CT (5 CDs, 169 towns)
ME (2 CDs, 433 towns)
MA (9 CDs, 351 towns)
NH (2 CDs, 234 towns)
NJ (12 CDs, 566 towns)
RI (2 CDs, 39 towns)
Perhaps require modest recognition of counties:
Two districts may split (share parts) of at most one county. This probably would result in better compactness, and would generally have reasonable road connectivity (you might have to dip across the district line near district boundaries, but not to great an extent.
Conceivably there could be a bonus for fewest county fragments, but there is a risk that in a state like New Hampshire, there might an extreme split that happens not to split any counties.
This might also be a better starting point for states to the west of New England which have a well-developed system of townships.