You don't catch a cold from being cold, you catch a cold from being inside all winter with other people. So you don't have to worry about him getting sick.
For really cold days, yeah I think I"m just going to have to put his coat on him really fast after we park if we're going to be walking along way to the entrance. If DH is driving with us somewhere, he'll usually drop us off/pick us up at the entrance. If he were doing that, I may not bother with a full-on coat for him, I'd just walk really fast. He'll wear a hat and mittens in the car, and it doesn't get insanely cold here, so I think we'll be ok. I don't know about people further north!