"You're the last thing on my mind" -- and that is present tense [are] and that suggests that very cool ambiguity you sense, and I do, too.
That reminds me of a girl who, a very long time ago, told me "you are the kind of guy I shall turn to if I have nowhere else to go".
By the way, the ambiguity of "the last time" also exists in German (das letzte Mal).