I am not sure that with with new connection can old contract be renewed but you will be provided with a new contract.
Rest you can verify with the Verizon employees.
you can upgrade the contract with them any time.
You have to contact any Verizon store and inform them you are out of your contract and want to start a new one with an upgraded phone.
If you are willing to sign on for a longer term wireless contract,
they will upgrade your cell phone no problem.
Just be ready to do so otherwise you may face some problem.
The longer you are ready to sign a contract the easy for your work at Verizon stores otherwise they create lot of nuisance and will take a long time to upgrade your phone.