The president will have to sign the existing senate bill first. Reconciliation is not a new bill. It is corrections to a previously passed bill.
The president wouldn't have to sign either bill, before the Congress passes both. In fact, Congress could pass the Reconciliation Bill before passing the original Senate bill, so long as the President signs the original bill before the reconciliation. That's the likely order of things and is how the House would assure that they wouldn't get stuck with just the original Senate bill.
They are two separate pieces of legislation.