I was born and raised in Florida. Generally speaking the farther south you travel the less likely that you will encounter the types of thunderstorms that prohibit flying.
In the south Florida region thunderstorms are generally localized. Occasionally there are thunderstorms without much vertical development that develop due to the nightly sea breeze along the costal areas that dissipate by mid am.
In the Central region thunderstorms start to develop in the mid pm but have dissapated by early evening.
The Cross City area is known as thunderstorm alley.
Lines of thunderstorms are more prevelant latter in the year. September and October are usually worse.
These lines usually develop in the gulf around Ft. Myers and stretch up towards Orlando to Daytona beach.
Another area for lines of thunderstorms would be from Columbus Ga. through Tallahassee down into the gulf.
A quick look at a weather chart in the mid am will give a lot of information about line development. A low with a trough extending from it will result in massive thunderstorm development along the trough as the temperature increases.
June in the south Florida area should provide one with excellent flying opportunities in the am and early pm.
Night cross countries are still a possibility if the thunderstorm activity was a result of land mass heating. These thunderstorms usually disspate when the heat source goes below the horizon.
If a thunderstorm has broken through the tropopause- watch out. These thunderstorms last well into the night and have lots of nasty side effects. Tornadoes, hail, etc.
One nice thing about flying in Florida is that there are plenty of airports to land at if the thunderstorm activity seems to be getting out of hand.
Finally, in all the years I lived and flew in Florida I never owned a raincoat. It wasn't neccessary because, as a general rule, one was pretty confident that if it was raining one could simply wait indoors for ten to twenty minutes and the rain would have stopped. The thunderstorm causing the rain in your area would have passed by.