Get some light leather conditioner, Lexol is fine and cheap enough, no need to overspend. Take the latigo home with you, let it warm up to room temp (warm leather is easier to manipulate the stiff, cold leather) sit in front of the tv and just work a small amount into the leather while bending, squeezing, stretching and flexing it for awhile. You can wear disposable very light weight latex gloves (drug store, $3) if you don’t want it on your hands…you want real thin so the warmth of your hands helps the pores of the leather stay open to absorb the conditioner as well as relax the leather…
Repeat once a week but don’t over do it, you don’t want it to get soggy, it’s the bending and flexing more then the conditioners…and avoid oil, that glops it up and gets it sticky plus can mark up your horses hair coat. Remember leather walked around on a cow, it’s alive.
There is a reason most long time riders have a love /hate relationship with new saddles, bridles, leather girths, boots, leather chaps and whatever. Patience and regular use make these things comfortable and allow the leather to regain its former flexibility and feel.
Dont like nylon because it never really softens up but have used it and very much understand why others prefer it. But I don’t mind the bit of extra work and sometimes shorter life span of the leather. YMMV.