What I do is is I check the Front Face First and then do an x' rotation once or twice to see the rest of the Edges. When there happens to be a lot of bad Edges, most of the time you can just do a Setup Move so that a F/B Move will solve 4 Edge's Orientation. Then I just remember those moves and look at where the other Bad Edges will end up.
For me, I try to strictly Blockbuild, but of course there is a case or two that may be better to just solve the cross piece first. Usually after the First Square you can have it offset with a L/L', R/R' Move depending on where the Square is so that it is possible to save 2 Moves from having to do a 3 Move insert, which changes it to a 1 Move insert after it's paired up, but that happens around half the time, called Openslotting.