This is a common situation, but not straightforward to solve in my opinion. I agree with the comment below, that if you start grouping things based on analysis, you are ignoring uncertainty. For example, if you group May and April together, but your estimate for both is quite uncertain, then this grouping may not be valid at all, especially for different years.
Answer, ignoring problems;
You can group after the first regression based on the coefficients for each month. Put similar coefficients together. If you have prior knowledge, you can group the months together based on that, but other than that, I think a regression would be the simplest way to figure out a good grouping.
Two other remarks:
- Not really your question, but, either way, this will actually create situations where your predictions drop or rise very suddenly when the month changes. If this is not realistic, you can look at including something like seasonal splines (see http://www.fromthebottomoftheheap.net/2014/05/09/modelling-seasonal-data-with-gam/).
- Probably a better and more satisfying answer can be given if you are a bit more clear about your modelling purposes, ie. what do you want with it, and why do you take these predictors, and why do you want to group the months like this?