3 Things:

1) I haven't checked if ethiad can interline bags with SQ on different ticket, that would save you time unless you really want to access checked bag during SIN.
2) if no interline, generally you can stay airside, visit a transfer desk, and get your bags tagged to SQ and boarding pass. No need to enter Singapore immigration (unless you want some local cuisine at the changi staff canteen/jewel).

3) if you still want to enter Singapore immigration.. That's fine, can visit lounge before entering Singapore using SQ boarding pass. Just make sure you exit Singapore (to go back airside) only once.


You can do the following (eg upon landing, spend 30 in at the airside transit desk to get stuff sorted out. Spend 1 hour lounge. Enter Singapore immigration and spend 3 hour at jewel. Exit Singapore to go airside and spend 1 hour at lounge. Board plane - overall only enter Singapore once and exit Singapore once).

You cannot do this multiple times (eg enter Singapore, spend 2 hour jewel, exit Singapore to go airside and spend 1 hour at lounge, then enter Singapore again to spend 1 hour at jewel. Then exit Singapore visit lounge and finally to board plane)