Replying to Editing Video with Flat Light Issues

Most of my video comes from my GoPro, and making edits with flat light is baffling me.  I'm just not sure how to optimize the shots so the quality isn't destroyed but the shots don't look too boring and grey.  Has anyone found some good fixes?

Also, on a sort of related note....when you make edits with both sunny and flat light shots, how do you optimize the video so that they meld together without too much distraction?

Any tips on what to play with would be sweet, the only decent results I've gotten have come from toying with contrast a bit.