"During the off-season"?... Frankly, that might be hard for potential employers to overlook. As you mention the Gulf, for PHI openings, click "Careers" at the top of this page
PHI, Inc - The Total Helicopter Company
Less familiar with Bristow, but looks like this link might be searchable:
Bristow - bristowgroup.com
I think this will work for ERA Helicopters:
https://erahelicopters.mua.hrdepartm...archType:quick
If you have sufficient night and an IFR rating, the EMS side of the industry might be a way to slowly build some turbine time. But now my fingers are cramping up... Good luck.