Gain from respinning the revolver in two-bullet roulette

A six-chamber revolver is loaded with $2$ bullets in random chambers and the cylinder is spun. You and a friend alternate pulls, and your friend goes first and survives. Before your pull you can either spin the cylinder again or leave it. By how much does spinning raise your survival probability compared with not spinning?

$1/15$. Without a spin, the next chamber is one of the $5$ unused chambers, $3$ of them empty: survival $\tfrac35$. With a spin, survival is $\tfrac46$. The difference is $\tfrac23-\tfrac35=\tfrac1{15}$.