If you want to immerse yourself more in general aviation, a good way is to be there, but focus less on being the pilot in the beginning. Everyone knows that if you're around the airport, you want to be flying, so be patient.
But, in the mean time, your very best formula is to pay for the modest amount of rental you can afford, and then spend time around the airport helping in a non flying role. Simply being at the airport puts you where the flying is happening, and the flying people are. That's what you want. You want to be known and liked by the people who own and fly 'planes. Your opportunity to ride along, even as just a passenger at some times, will demonstrate you're worthy of their investment in you. You'll be along, and learn some things. The pilots who get to know you will know that you've been a [albeit junior] member of the group, and are worthy of the trust.
Offer to pump fuel, wash windshields, carry bags, and scrub dirty aircraft bellies. When you see an interesting aircraft type, go and chat with the pilot about it - pilots love to talk about their 'planes! Your best opportunities in aviation happen when you're at the airport! The piloting will come in time, demonstrate patience....