Own a modern player so will chime in here. In mine I am running a 59 Custom hybrid bridge, Alnico II pro tele neck and a standard SSL2 for the middle. The tele neck pickup is gong to be RW/RP so use a standard middle pickup to get hum canceling in the middle also will set you up as humbucking with a standard split on the humbucker.
While I really like the JB in the right guitar and have a few I would highly recommend the 59/Custom in a pine body modern player in particular for the type tones you are looking for. Less compressed and much more clarity + has amazing harmonics in one of thee guitars. The Alnico II neck has a soft warm singing tone that should really work also and the SSL2 is a non staggered vintage toned clean clear pickup that will give you a solid strat middle tone + those strat like 2 and 4 positions with a split 59/Custom or the Alnico II tele. Love this set up in my MP so good luck!!
59/Custom hybrid in a Tele