If you live in a place where the winter is a lot colder and drier than the rest of the year (e.g. the subarctic), it makes sense to rehang your ax in the winter, if you have a choice. The reason is that the wood of the handle will have less moisture in it during the winter, in comparison to the rest of the year, so it will have shrunk slightly. If you fit a new handle in the winter, the fit is likely to be even more solid the rest of the year. If you fit a new handle during a warmer and more humid time of the year, the axe head would be more likely to loosen up when the wood loses moisture and shrinks in the winter.
If you’re in a situation where your axe head is a tad loose, but the handle doesn’t need to be replaced, soak the eye in linseed oil overnight. The oil will swell the wood, thus tightening up the fit. What’s nice about using linseed oil is that it will remain in the wood and harden. I think it’s a better solution than swelling the handle with water (which, granted, also tightens up the fit, but will eventually lead to weakening of the wood if done repeatedly).