Tennessee courts generally look disfavorably on non-compete agreements. The Tennessee legislature allows non-compete clauses that are less than two years and limited to a 10 mile radius. However, sometimes courts will completely nullify a non-compete clause for various reasons.
Quick answer to your question: maybe. This is one of those situations where the language of the contract is really important. Without seeing the contract, my best response to you is that sometimes these clauses are enforceable, sometimes they are not. There is no quick answer that I could give you that would solve this for you without seeing the contract, other than asking the hospital not to enforce it, which is always an option.
If the hospital agrees not to enforce the clause, then you are fine. I know it may sound silly, but many people believe that just because a clause is in a contract that it must legally be enforced. This is not true. If both sides agree not to enforce it, then it is as if it never existed. But I'm guessing that this is not an option for you.
My best advice is to have a lawyer look at the language of the contract and give his or her opinion. This can be done very quickly and at little expense. If you find a lawyer that wants to take a long time and charge a lot to answer this question, then go find another one. Fighting this clause is the thing that might cost some money, but getting an opinion about possible outcomes is not. Good luck!
You're likely looking at some sort of battle in Chancery Court downtown to get the non-compete lifted or modified. As much as non-competes are disfavored in TN, it is quite a different thing to deal with them in a health care setting in Shelby County -- lots and lots of variables at play. Feel free to call or write off line to discuss further.
The foregoing is not legal advice nor is it in any manner whatsoever meant to create or impute an attorney/client relationship.