I don't really do anything different.
If your chemicals are good going into the rain, then you should be fine.
Spend time reading troublefreepool.com about chemicals
The issues you should be aware of:
1. If your chlorine levels are low depending on your CYA levels, then it can dilute the chlorine enough that you get green pool. This can definitely happen if using pucks. I run liquid chlorine, so it's never a problem for me, unless I screw up my chlorine feed
2. Other chemical values can also go down due to rain including CYA, TA, CH, CL. Those aren't a big immediate impact. Just good to know and track over time
3. Rain can cause a bunch of junk to get in the pool and end up in skimmers. Check those to make sure they aren't crazy full