Primer by itself will not prevent the stain from travelling up the paint. You need a primer with a sealer and stain blocker. This gives it a non-mixing medium that the stain can’t mix with, thus preventing bleed through. Do a light sand and use the stain blocker primer that should do the trick.
Doing a second coat of such a primer is preventative but not a fixed rule. Sometimes one coat is enough. - redracer01