Is the initial non-compete clause still valid if I was rehired by a company?
First, courts are loathe to enforce non-compete clauses because doing so may prevent people from obtaining gainful employment. In addition, to be enforceable, courts want to see narrowly tailored clauses (not "you can't work anywhere in this industry"), particularly if you did not hold a highly sensitive position at the company that granted you access to vital company confidential information.
Part of the "narrow tailoring" means taking a careful look at the time frame of the non-compete. The longer the time frame, the greater the justification the company will need to enforce it.
To get a clear read on your situation, consult with an employment attorney to review the actual language of the severance agreement.