Have you run the YTF (YIG tuned filter) calibration? I have an 8596E but haven't used or cal'ed it in a while, so am not sure if your frequency cal is the same thing.
The YIG tuned filter can have some serious degradation of alignment from heat and age and not generally adjustable by most humans. If you get the YTF to pass, then there's hope for the unit.
To experiment (yes, play) with higher frequencies a few years ago, I've used a Hitite (now Analog Devices) frequency multiplier eval board (4x input frequency, but $500 for eval board, chip is around $40 or so), inexpensive tuned YIG oscillator (ebay) and some $7 , 10GHz location sensors (fun to play with, work on Doppler principle).
But there are some USB, GHz synthesizers out there now like this one for $249 (
https://windfreaktech.com/product/usb-rf-signal-generator/ ) which goes to 4.4GHz. Not sure about output leveling or accuracy.