Hi Travis -
One way that worked for us, was to register a new sender domain thats related with the original domain, and send a notification to all gmail users to check their promotions/social tab and drag the newsletter email saving the setting. This depends on your email sending structure, do you own a dedicated IP or use a shared ESP.
We saw a 8-10% reactive gmail customers, but it all depends on the quality of your list.
Is not a perfect solution but is a start