I would implement it with a cash rebate to everyone in America for the average number of gallons used by an American in a year * $10 and call it the "green rebate" or something like that and make it clear that it paid for by a gas/car emissions tax.
Australia tried something similar with a carbon tax a few years ago - not for fuel, but for carbon emissions generally. The policy was very well designed and ensured that lower income earners were compensated for any increase in costs to the extent that they had a net benefit, and in the short time it was operating, was successful at reducing emissions.
Unfortunately for us (and the world) this was too complicated for many Australians to understand and the political right exploited that to tell a scary story about a 'new tax', leading to their election and the removal of the scheme.
While economists (and rational thinkers) generally love carbon pricing schemes, they have been pretty unsuccessful politically because people are generally too stupid to understand them and cynical politicians in bed with the fossil fuel industry are happy to play to that.
Edit: For clarification, if it's the type of rebate typical in the US, that means I get it later and I can't afford the fuel now. If it's a rebate that's done at the time of purchase, then it wouldn't seem to make much difference, on average.
Let's say it's annual (or if they have the capacity, monthly) and paid at the start of the passage of the law, so you get the money first before the gas price increase.
It's not done at time of purchase because it is not a gas subsidy.
The average American would be paying $0.