If you catch a coon, the whole area within the catch circle smells like coon. Not just the trap. Should have no problems setting the same trap back.
Don't worry about touching the trap with your bare hands. I set bare handed when it is warm enough not to wear gloves. You will have more problems by allowing lure to get on your traps or improperly bedding them. Both of those reasons can lead to dug traps.
Also, in my area, some coyotes refuse to enter the catch circle. Always have a fresh set at that location. You really need to have at least two traps at each location anyways. If the spot was good enough for one set, it is good enough for two or three.
If you are worried about the condition of the trap, just take it to the car wash and hose away. When you start seeing some rust then you might consider dye and wax.
Randy