Alternatively, find the digit that, in its row/column/square/whatever, has the minimum number of choices available, and try each of them.

Ideally, combine the two and use Knuth’s dancing links algorithm (https://en.wikipedia.org/wiki/Dancing_Links). It, at every step, picks the one of those two approaches that has the lower number of possibilities, typically getting a lower branching factor, without spending much more time at each step.