He got his new costumes as an actual plot point. Past a certain point in his show, he got re-upgraded and gained his Shin Kamen Rider form and powers.
Saying that the upgrades were "actual plot points" is kind debatable for the original series. They're acknowledged by official material and have official names and, in the show itself, Rider 1's abilities clearly change with the different looks. However, they don't bother actually explaining, in the series itself, how he gets them, only in external material.
In the show itself, episode 53 just starts with a new Shocker attack and then Hongo makes a transformation pose, something he never had before in the tv show, and then suddenly became the new Rider 1. He also gets another power up a few episodes later ("Rider Power!" by pressing a button on the side of his belt), although that doesn't change his look, aside from making the helmet lights blink, but that's just used twice.
Skyrider's upgrade and Stronger's form change were actual plot points though.
Ok, so V3, ZX, Shin & ZO are the only Riders that since their birth they have not undergone any kind of change at all.
V3's "changes" were just him learning more about his own abilities. IIRC, one of V3's secrets was some kind of temporary power up that let him just tank the attacks of one of Destron's monsters and easily finish it off, although it didn't change his appearance.
X, while he didn't get a form change, he got an upgrade, it changed his transformation and gave him more power to do new finishers. It's pretty much the same as Stronger's just without any visual change.
In his show itself, he also became more of a brawler after getting that upgrade, using the Ridol less than before and going more with punches and kicks. Although later appearances pretty much reverted him to "the guy with the stick", and usually even ignore the Ridol's other modes (like the fencing sword).