The other way is to take the Skytrain from T1/Jewel to T2 (which is the closest skytrain to Jewel), then pass through immigration in T2 and then take Skytrain from F end of T2 to A end of T3, which is quite close to the SKL/F lounge entrance.
But if you must, you could always download the Grab app and pick up a Grab in the basement of Jewel/T1 to take you to the First Class Reception in T3
At the moment quoting $6 for that ride, but could be higher during peak hours.