I finally found the book and thus the series I’d been attempting to re-discover!

It is the Joe Ledger series by Jonathan Maberry. I’ve only read the fifth book in the series (Extinction Machine) so far but will likely read all the rest of them now.

I had unknowingly read an entirely different series (the Rot & Ruin one) by this same author between reading Extinction Machine and then attempting to find it again. (I read a lot of books.) But after all these months of searching off and on I am glad to have discovered it again.