I dont know how many hours your planning on flying a day and how long your planning on staying at each stop, for the flight your planning you should be fine but i would check on Basair's policy regarding long term hire and min hrs.
As far as I am aware on everything but there low use higher end aircraft they have minimum hrs flown per day if you take an aircraft for long periods as they dont want to miss out on the money that plane could be making, but check this to make sure its still true.
If you dont want to be constantly worrying about this and have a more relaxed fly I would check with a few places and maybe consider the flying club.
For a long hire they will make you pay a chunk in advance but they tend to be more relaxed about hrs flown and since they own most of their aircraft they tend to be in slightly better condition anyway.
Hope this helps, if you want to ask more questions please feel free to PM me
Zepth