my 2013 auto has a weird second gear also, not sure if it's the torque converter slipping then locking up or what. It seems like it shifts 2 times in 2nd to 3rd gear. As RPM climb in 2nd it drops about 200 rpm at a certain point then climbs again before finally firmly shifting into 3rd. Kinda weird to explain. Sounds different than what you're describing.
I'm not sure if any aftermarket tuners support changing shift points on the 12's and newer auto transmission. My Diablo Sport Trinity doesnt do shit for adjusting shift points even tho the website says it does. DS says they can't talk to that transmission. Not sure if that's true for ProCal or any of the others tho, may wanna make for sure before droppin the $$ for a tuner. And if someone on here says that a tuner helped out their shift points, make sure their jeep is a 12 or newerwith the 5spd auto. Some tuners may have worked on the older 4spd auto that allowed changing shift points, don't know, but that doesnt necessarily mean it's gonna work on yours.
'64 CJ-5, '74 Bronco, '79 CJ-7, '84 CJ-7, '92 XJ, '96 XJ, '99 XJ, '01 TJ, '01 XJ, 2013 JKU.
2013 Commando Green JKU
35x12.50/17 Falken Rocky Moutain ATS
2.5 RK X-Factor
17" Black Rock Vipers
Cut Stock Flares
Diablo Sport Trinity (garbage)
Sleeve, Truss, ARB, Chromo D30