That's a good question and as a lifelong Rita fan, this is what I have come to understand about her beginnings as an actress, which may relate to what you're asking.
Old Hollywood (and current, lets be honest here) was extremely racist. The likes of Dolores Del Rio, Carmen Miranda, Anna May Wong, etc. were always typecast and that restricted their access to many roles, which many white actresses could take on.
From a young age (16, I think) Rita was making films, usually bad B-movies and in them, she has her natural black hair, typecast as foreign characters due to this (Egyptian, Spanish, etc.). She used her real name at the time too, 'Rita Cansino', which sounds Spanish. Both aspects held her back in trying to establish more of a career (according to her father and manager at least). However, what people don't realise about Rita is that it was her manager (and later, 1st husband), Ed Judson and her father, Eduardo Cansino, that wanted her to be a star. Rita was never attracted to the Hollywood way of life and judging by the descriptions of her character (EXTREMELY shy, docile, agreeable, etc.), she was very easily pushed around and controlled by these two figures in her life.
Ed Judson told her father that in order for Rita to make a career, she needed to move away from her natural image and birth name. This was the birth of 'Rita Hayworth'. Hair electrolysis, red hair dye, a name change, weight loss program and a new studio to back her seemed to have did the trick. From this, she was able to get other roles besides always being typecast.
So basically what I'm trying to say is that it may be just because Rita had help and advice on how to go down a different avenue in her acting career. She still acted in non-white roles but her new look allowed her to tackle more than she was used to, much to the delight of Ed and Eduardo.
There maybe was no stigma to being Spanish in hollywood but at the same time, she still would have continued to have been typecast due to her looks and Hollywood still does it a lot of the time.