I think that we can discount Lt. S Molinari. Anton Molinari, however, sounds like it could be your man. He would have been of Italian decent, as was another famous Molinari..Karl-Theodor, so not unusual.
The date on the jacket is 1937 so he was in the Luftwaffe at an early stage at the rank of Lt...checked to see if he appears in the Legion Condor lists...he does not. So further research into Anton, and the squadrons he was in, is the best bet. The thing about the WW2 German military is that they formed, and reformed, groups/units for specific objectives which makes research a little difficult at times, but having some of Molinari's units is a good place to start as documentation usually gives what a particular unit was formed from, when and the names of the officers.
Btw...any chance of posting a pic of the jacket as it may hold clues to awards...by way of loops/holes etc.