I'm totally not sure that else branch is appropiate here for 'i == w2.
Even since we have i > w for this m this logic isn't clear for me.
What about simple replacement i <= w for loop condition with classical i < w?
It could be enough and cleaner.

I think the else branch will be OK, since all it does is advance the index, and it is intended and expected that this could potentially advance it past the end of the string, in which case the for loop will terminate.

We can't just replace the the loop condition with i < w since the w here is the last valid index that the pattern could appear at, and needs to be checked. Otherwise we would miss valid matches, and indeed such a change breaks a large number of unit tests.

Note the conditionals that were changed are miss conditions, i.e. the algorithm has determined the character at the index cannot be part of the pattern at this location in the string. The conditionals modified are checking whether the following character could potentially be part of a pattern hit, so as to determine whether to skip it entirely by advancing the full length of the pattern or only as much as possible while still considering it as a valid potential hit. In the case where we are at the end of the buffer it doesn't actually matter which branch we take, since either way it will advance past it and terminate. We just need to avoid reading the invalid following character when it doesn't exist.

Maybe it's better to rewrite condition as i+1 <= w? It seems to be more obvious way of checking for me.
It's very similar to "for" condition, but for another argument and before direct using of i+1 as index.

... the w here is the last valid index that the pattern could appear at

IMO, rewrited condition slightly reduces cognitive load.