Rule for "Whomever" or "Whoever" following a preposition

My rusting brain seems to recall a simple test for deciding which case to use in thie following instance. I think it goes something like this: If the pronoun (Whoever/Whomever) is the subject of the dependent clause, then the proper form is "Whoever." E.g., First prize went to whoever performed best.

But if the pronoun is the object of the dependent clause, the correct form is "Whomever," as in "The girl said she would deliver the note to whomever it named.

Am I correct? If not, why? And if I am, is there a better way of expressing the rule?

Thank you.