Here's an observation from my line: A while back I had my "skinner" riding with me. He's a trapper also, and constantly on my butt about bloody heads, etc.
Long story short, I head-shot a fox, took it out of the trap and began the remake, Ole Skinner grabs the carcass, carries it across the 2-track (15-20 feet), holding it by it's back legs, until I'm ready to go. We got a little snow that night, just enough to track. The next morning, I tracked where a fox made one trip past that set, and walked all over the place where the previous fox had bled out.
If I end up with a puddle of blood in the catch circle, I'll move it out of my work zone. If it's behind or off to the side of the trap, I cover it with a scoop of dirt or grass. The only reason I cover it, is to keep crows/hawks, etc. from messing with it. I figure if an animal wants to sniff around blood, I'll keep it near the set.