He's not got much in the way of experience in being a leader despite him being 'Captain' Han Solo for more than a year already. A little more experience giving someone other than Chewbacca directions is probably good for him. And her a more forgiving presence, perhaps.

"I don't think that makes a lot of sense to be honest." He says, as Han pushes the tavern door open with his free hand. "You should keep the jacket on. We're going somewhere that I'll have plenty of stuff to change into if I get wet."

That's the best reason he can come up with on the spot for justifying her keeping the jacket. To be fair, it's not a bad point to have made either. Mostly it's a sense of chivalry that's in him that wants to see her stay warm while he toughs it out. It must have been raining the whole hour or so they were in the bar because the temperature has noticeably dropped.