Yes. Find out what leaving your current provider may cost you before helloBill sets up your new one.
If you're moving home
Contact your current provider yourself. Tell them your move-out date and ask if there's an exit fee if you change to a new provider for your new address. If they can serve your new address, many will offer to move the service with you and may charge an exit fee if you're still within your minimum term but want to change provider. If they can't serve your new address, ask them to waive it. A waiver isn't guaranteed, so get any agreement in writing.
If you're staying at the same address
Your new provider arranges the transfer, so don't cancel the old service separately unless they tell you to as part of that process. Once the switch request is sent, your current provider must tell you about any early termination charges before the switch goes ahead, and you can still stop it. Before you start, check your current contract for an exit fee: outside your minimum term there often isn't one, and inside it you may be asked to pay for the remainder.