Absolutely, the worst offenders might not be as bad as we think 
As @X0rr0 previously pointed out, you can switch from cancelling 2s (when w and u have a shared prefix) to cancelling 3s (when they have a shared suffix), or you can mix (when they have both).
@oros mentions more powerful cases of 2- and 3-canceling, which don’t necessarily have anything to do with shared prefixes or suffixes.
These often feel kinda “unpredictable” or “accidental” to me, or at least even harder to model that what we already have in front of us. But it’s great to realize there is more headroom if it comes to a crunch.
One of my favorite cases of “accidental” canceling is ruling out a particular circuit, here with a twist on the single-member method:
If circuit v = 1111111100000 is an integer cycle, then its bottom member \cfrac{3^8 - 2^8}{2^{13} - 3^8} is an integer.
In that case, 31 \cfrac{3^8 - 2^8}{2^{13} - 3^8} + 206 is also an integer.
But 31 \cfrac{3^8 - 2^8}{2^{13} - 3^8} + 206 \cfrac{2^{13} - 3^8}{2^{13} - 3^8} happens to equal \cfrac{3^{12}}{2^{13} - 3^8}, which is not an integer.