Premium economy sold out, how to maximise chance of getting an upgrade if a place becomes available

Flying in the next couple of weeks London to USA and back; will be fully refundable tickets. At present, no availability is showing for premium economy on the transatlantic flight back to the UK - if one became available I’m happy to pay the difference to go up. If I book with premium economy out, economy back, will I have the opportunity to upgrade if a seat becomes available? Any suggestions on when best to look - I’m guessing mostly if people fly standby a few hours earlier?