I want to highly recommend the Jura for hiking out of several reasons. Firstly you will encounter very different highlights which makes for varied hiking. There is the Doubs valley, the lovely mountain pastures and the breathtaking views onto Lake Geneva and the Alps. Secondly the Jura had an alpine flavour without really being that high altitude. This makes it an ideal destination for people who are looking for alpine hiking when it is too early or too late for the Alps themselves. Of course the Reculet is not Mont Blanc, but the Jura is an interesting middle between real alpine hiking and just rolling hills. In mid June there was not a bit of snow left and I guess they are hikeable by early May. Plus there are a lot of snow shoe trails.
But what makes then most interesting is the possibility of doing various loop hikes as most of the time the GTJ (Grand Traversee de Jura) and its variants are paralleling each other and interconnect. You could fly into Geneva, take one of the almost hourly trains to Bellegarde for just 6,70 EUR and start a loop hike from there. Or just hike from Bellegarde to Culoz in 2 days from where there are several trains back to Geneva via Bellegarde. The ascent/descent Bellegarde/Reculet is very steep and tough, whereas the climb out of Culoz onto the Jura range is long but gentle.
I found wild camping relatively easy, but there are also plenty of refuges and gites. Try to get the relevant topoguide for the GTJ that describes not only the GTJ itself but ALL the variants and possible accommodation.Waymarking in the Jura was much better than in the Vosges and on the GTJ you will not need a GPS, only on the less hiked variants waymarking could be a bit better. There aren't many hikers in the French Jura, it is still a bit of a secret. The only problem I encountered was the lack of water on the crest, but with a little bit of planning you can easily manage. I recommend a water capacity of 4 liters. Now go and enjoy yourself !