Pedro, I've had four basses over the years - 2 Fender Precisions, a Yamaha (a la Van Halen's Michael Anthony), and currently a Fender Jazz.
The first one I owned was a Made in Japan Precision that I kept for well over 20 years; it was a joy to play, but felt like it was made of balsa - so light was the construction, coupled with the old style machine heads, that it was actually neck-heavy, and only felt comfortable playing sat down.
I traded that for the Yamaha a few years ago. It had active pups on it and sounded great, but the neck was just too slow for me, it felt like treacle compared to the Fender.
After about a year, I traded that against a used USA-built Precision. As well as the obvious improvements in construction over the MIJ model, it sounded great (as one would expect), but it never felt as comfortable to play as the other Fender.
My epiphany moment came when I was down at my local Guitar store, and I had a play on a Fender Jazz. Similar to yourself, I suffer from very short fingers, and once I started playing around on the Jazz's neck, I was hooked. It was so much easier to play, and I decided the and there I had to have one. As this was a model from the 70's that he wanted well over £2000 for, I opted for the cheaper option and bought a shiny new US Standard Jazz.
My advice (and if I'd followed it myself, I would have gone straight from the MIJ Precision to the US Jazz!) would be to actually get some hands-on time with both a Precision and a Jazz, and see for yourself the differences. The difference in neck radius doesn't seem much 'on paper', but when you need every little bit of help in wrapping your hand around the fretboard, that difference is substantial.