Tadas,
It sounds like you're experiencing a bug that was introduced by Facebook last year. They changed how they notify JFBConnect of an automatic login. We're looking into a solution to the problem, but haven't found exactly what we're looking for yet.
The 4.1.2 release of JFBConnect is coming out tomorrow or Wednesday. After that, this is going to be a higher priority bug for us to figure out. Until then, there's not a better way to make auto-login work, unfortunately.
Hope that helps,
Alex