Short (tldr) answer:
You may not want to book the 7:50AM flight to Phuket (HKT), mainly because of potential unreliable arrival times on UA1.
Long but anally complete response:
The answer to your question is unclear because Scoot is a low cost carrier, thus amenities and practices we accept for granted on full service legacy airlines (such as UA) may not hold. In any case the issue is about Scoot's check in practices. You are purchasing two distinct tickets with two separate PNRs so no interlining is possible (it wouldn't be anyway given Scoot is low cost).
Scoot itself is clear as mud on transits.
http://www.flyscoot.com/en/fly-scoot...changi-airport
Point 1 says to stay in the transit area and then makes reference to their Scoot thru service, while point 4 says you have to legally enter Singapore.
Sooge_sj is correct in describing the transfer desk facility on airside but with due respect to him/her, it is unclear if non Scoot-thru passengers can use this. One sticking point is that there is a charge to bring on board carry on baggage more than 10kg; I don't see why the overweight onboard bag fee can't be paid at the transfer desk but crazier things have happened (so don't take anything for granted). The good news is that if you have to go to the check in counter at Departures, Singapore Immigration and Customs is incredibly quick; it's possible to be from jetway to exiting Customs in 20 minutes. YMMV of course but 30 minutes can be safely assumed. Entry into airside (Immigration formalities) is just as quick although the actual check in might be slowish.
What you should do on arrival is to head directly to the transfer desk and see if they can check you in (it's either Transfer Desk E or F depending which terminal finger your gate is located). If so, great. If not, go through Singapore Immigration and Customs, then proceed upstairs to Departures to check in there (row 10).
T2 map for transfer desk. See transfer E next to gate E1 or transfer F next to F30.
http://www.changiairport.com/en/maps.html#map/T2L2/
check in row
http://www.changiairport.com/en/flight/airlines.html
You may not want to book the morning Scoot flight you're thinking of but the evening flight at 6:25PM given UA1 might be delayed arriving in SIN. The arrival today (August 1 this end) was at 7:19AM. These delays are often due to hiccups departing from the US end. It would make sense for you to research the history of UA1's arrival times into SIN to help you decide which of the two Scoot flights you want to book. (BTW if you book the early evening flight, Changi Airport offers a three hour Singapore city tour that might be interesting if you have never been to Singapore before. This will help kill some time.) Or you might think about flying Thai AirAsia (FD) or Jetstar (3K) instead especially if their flight timings are more convenient.