Non-Straw Hat characters, whether realized at the time or in hindsight in regards to the arcs they first appeared in, that you feel have gone on to have the most relevancy within the series. They don't even have to have necessarily had large roles within the arcs they first appeared in, so long as they appeared in the first place. My choices would be:
Romance Dawn Arc - Shanks
Orange Town Arc - Silvers Rayleigh
Syrup Village Arc - Jango
Baratie Arc - Dracule Mihawk
Arlong Park Arc - Hatchan
Loguetown Arc - Monkey D. Dragon
Reverse Mountain Arc - Nefertari Vivi
Whiskey Peak Arc - Igaram
Little Garden Arc - Crocodile
Drum Island Arc - Portgas D. Ace
Arabasta Arc - Daz Bonez
Jaya Arc - Marshall D. Teach
Skypiea Arc - Enel
Long Ring Long Land Arc - Kuzan
Water 7 Arc - Rob Lucci
Enies Lobby Arc - Sakazuki
Post-Enies Lobby Arc - Thatch
Thriller Bark Arc - Shimotsuki Ryuma
Sabaody Archipelago Arc - Trafalgar D. Water Law
Amazon Lily Arc - Boa Hancock
Impel Down Arc - Emporio Ivankov
Marineford Arc - Izou
Post-War Arc - Sabo
Return to Sabaody Arc - Caribou
Fishman Island Arc - Charlotte Linlin
Punk Hazard Arc - Kinemon
Dressrosa Arc - Kaidou
Zou Arc - Pedro
Whole Cake Island Arc - Charlotte Katakuri
Levely Arc - Im
Wano Country Arc - Kozuki Oden