Did you apply sideways to the school of social work at your present school? That would probably almost automatically take you if your grades thus far are okay; they know you are a "sure" thing, no thought required

If you are trying for other schools, they might think similarly, if they don't I'd go sideways and take a few of the classes and then try to "upgrade" to a better school if I thought that was needed. What school one goes to seems really important when you are going to school but after you get out and start actually practicing, the import of the school becomes much less. If, in practice, you are a really good social worker, it won't matter what school you went to, as long as it's a respected degree-granting institution.