CH's first sentence is correct for self-centering (almost all 3-jaw and a few 4-jaw) solid-jaw chucks. But not for at least most 3-jaw with 2-piece jaws. There are apparently two standards, American Standard Tongue & Groove Jaws and something roughly equivalent but made to metric dimensions. The jaw dimensions vary according to the chuck diameter (and 8.25" actual diameter appears to be standard on 8" chucks). So you only need to confirm that you chuck has American Standard Tongue & Groove Jaws and look for those to fit an 8" chuck. For some strange reason, there is no mention of them in Machinery's Handbook. But they appear to come in several nose shapes. Prices are all over the map. But I wouldn't buy the cheapest that I could find.
You never posted a photo of your chuck so I don't know why its minimum diameter for clamping is about 1".
EDIT: If you do buy a set similar to those in your second photo, don't forget to re-center the chuck after you install them.