You could do something like add two filters:
In future after 5 months
In future 6 months
Although that won't be as accurate, it's probably good enough for your use case.
If you really want to make sure it covers only the full month of June like you said, I would just create 12 smart campaigns with the two filters like:
on or after "6/1/16"
on or before "6/30/16"
Downside there is you would have to redo it each year.
I've done what Justin has suggested before, and that works, especially when you have the campaign scheduled to run on the 1st of every month—then you're controlling the date filters, so you're not hitting dates that are in the middle of the months. That way you have the smart campaign scheduled to run every month, and you just have the same "in future after 5 months" AND "in future 6 months" filter on the date you're working off of.
We have run into a problem before where you can't have wait steps in those scheduled campaigns, but you can be clever with using batch smart campaigns that lead to "request campaign" and convert the flow into trigger-based with wait steps.
In addition to these suggestions, there are several related threads if you look up "expiration" or "renewal date"
I've spent a lot of time on this issue as all our renewal notices currently run through Marketo . My best workaround was to create a formula in SFDC to calculate a specific day before renewal (90, 120, etc.). It's worth doing if this is something you will continue to have in your programs for the entire year. I set up a weekly send for X days before renewal equals last 7 days.