Hi, I walked the Camino in three parts over three years whilst working then in completion when I finished work, always with a early/mid June start.
I would definitely go with a St. Jean start as I think you will have a much more satisfying time having started at St. Jean.
It is relatively simple to either get back to St. Jean or onwards to Santiago by public transport, bus is best, from most towns along the walk, check the bus services online.
I wish you Beun Camino
AlanS