Honestly..... I wouldnt do it. I think there are "other fish in the sea", and if you really arent happy with your current stylist, you can shop around and find a stylist at another salon.
Salons are just so.... political sometimes, you know? I realize that "business is business", but for me its also about etiquette and tact.
I just think that your former stylist will be hurt/offended.... it will definitely impact her relationship with the new stylist.... your relationship with her (not that you necessarily care)..... there will be tension in the salon when you come in.... so on and so forth. *shrug* I would personally feel too uncomfortable, and I dont think its worth it.
If you MUST stay with this new stylist, i would personally do as another young lady suggested and see her on the days that your former stylist is off. I know thats hella passive aggressive but.... i just dont like being in uncomfortable situations. And it would be pretty awkward for me.