Two things can happen here actually:
1) JAX Center can hand you off to TLH to continue flight following. They do this by automated means (radar) and the controller at JAX will tell you to contact TLH and give the frequency. For this to happen, you would obviously have to be flying at an altitude below TLH's ceiling.
2) Depending on traffic and they altitude you are flying, etc., the controller at JAX may call up the approach control facility at TLH and conduct a point out if you are just transitioning the airspace (not landing at an airport located in TLHs airspace). What a point out does is tells the TLH controller that your aircraft wants to enter their airspace but the JAX controller does not want to hand over communications to the TLH controller (possibly because when you exit TLHs airspace, you would be talking to him again). From there, the TLH controller either approves or denies the point out. If approved, nothing happens on the pilot end and you continue talking to the JAX controller. If denied, the only other options left are to do a handoff to TLH or the JAX controller would vector you around TLH airspace. Most likely, they would just hand you off to TLH.
Otherwise, follow the controller's instructions. If they tell you to contact TLH to continue flight following, do so (the TLH controller should know that you are getting flight following so you won't have to tell them everything again). Otherwise, the JAX controller may just get a point out and if you notice that you are in TLH's Class C and you are still talking to JAX center, then that is most likely what happened.