I read that the stars orbiting around Sagittarius A* (a.k.a the supermassive black hole located at the heart of the Milky Way) move very fast (many times faster than our Sun moves across the galaxy), and it is believed they have enough speed to wander off. My question is, is any exception to this rule, where somehow some stars may break a stellar speed limit not due to a black hole in their vicinity?
No, black holes are not the only cause of HVSs, although it is thought to be the most common mechanism.
Hyper velocity stars are believed to be caused
when binary stars come close enough to a supermassive black hole for one of the pair to be captured while the other star is ejected at high velocity. This appears to the main mechanism for HVSs. See for example this Harvard Smithsonian Center for Astrophysics page, and this University of Utah page.
when one star of a binary goes super nova, the second star may be ejected at high velocity. See US 708: Hypervelocity Star Ejected by Supernova Breaks Galactic Speed Record and How do Hypervelocity Stars End up Breaking The Speed Limit?.
There may be other causes, since not all HVSs seem to be explained by these mechanisms. Perhaps some of them are not from this galaxy at all - they may be just passing through.