The $6,000 is for direct contributions, there's no limit on the amount to roll over. If you have a 403b after leaving an employer, you can keep it there (if >$5,000, otherwise your old company can force a cash-out to transfer), roll it to your own IRA, or transfer it over into your new work's plan if they allow it.

So if you had $10,000 in a 403b, you can keep it, roll it to an IRA, or if your new job allows move it to them. The money stays pretax, there's no tax added to your 1040, and no penalties. A small wrinkle if the transfer is done via paper check, but if that happens just ask us... within 60 days of getting the check.