I suggested the Fender American Original 60's Jazz even though it doesn't have the same neck that the 60's Jazz Basses had. No idea why Fender insists upon calling this "Original 60's Jazz" when it uses the modern 9.5" radius on the neck instead of the 7.25" radius that these 60's Jazz Basses came with.
I had a 1965 Jazz that I bought in 1966 and among other things toured with it for three years in the early 70's playing 5-nights a week, 50-weeks a year. So I got to know that neck really well. My '65 was stolen in 1976 and it wasn't until 1985 that I found another MIA Jazz with the same 60's slim C-shape neck with 7.25" radius. I bought that and have been playing it for the last 35 years.
So I'm an extreme case. And if you haven't already developed a preference for a 7.25" radius Jazz neck, the flatter 9.5" radius will probably be fine for you. The Slim C-shape neck is a wonderful neck and the easiest of all their shapes and profiles to play. Hard to go wrong with the sound of those vintage pickups too. The one thing that drives me a little nuts on that bass cosmetically though is the "grounding strap" running from the bridge pickup to the bridge...there never was such a thing, at least not up through 1965. They think it looks cool, I don't think so. But it plays nicely and sounds good.
I like a number of things about their Original 70's Jazz better, such as no "grounding strap" and the 70's pickup spacing - 4" pickup spacing measured center to center. The 60's spacing is 3.6" center to center. The 70's spacing is a slightly brighter over all sound because that pickup is located closer to the bridge, and it growls very easily if you dig into the strings over the bridge pick-up. The 60's one still growls, just not as easily and without quite as much bite.
I am not a fan of white bar markers on the neck of the 70's bass though I do really like the maple neck. It just needs some contrast. White on Rosewood is good but IMHO you need black on maple. The thing that absolutely kills any chance of the 70's Jazz being preferred though is that it uses that U-shaped neck. The only neck possibly worse than that was the deep-U shaped neck on the 1966 Fender Jazz, which was the first Jazz with design changes made after CBS bought them in 1965. It's absolutely a terrible neck and of course uses that flatter 9.5" radius that modern Fender basses are so stuck on. It is a nice and narrow neck like Jazz necks are supposed to be but with that U-shape, it's fatter than some P-bass necks I've played. It's abominable.
The Fender American Professional II is a pretty nice choice as it has that Slim C-shape neck on it albeit with the typical 9.5" radius. I like the maple neck on it. Not sure the hi-mass bridge really does anything to the sustain but it has a nice look on it too. My maple neck Jazz has the best sustain I've heard on a Jazz and it just has that standard "bent-bridge" which is definitely not hi-mass. I do like the idea of the graphite rods neck reinforcement though to minimize neck reaction to weather and elevation changes. Not sure I'd want the new modern pickups although they do sound good in Fender's video.
So here's what you do. You get the 70's body with the 70's pickup spacing and actually have Fender put a Slim C-shape neck with a 7.25" radius on it and call it done. By the way, the MIM Fenders have the 7.25" radius on their re-issues and if I recall correctly that have the "original" pickups too.
