Sam's Leadership Hero card says: "Response: After you engage an enemy with a higher engagement cost than your threat, ready Sam Gamgee. He gets +1 [Willpower], +1 [Attack], and +1 [Defense] until the end of the round." Here's my question:

If Sam Gamgee is already ready, and is then engaged, does he still get the +1?

Yes. The subtle difference between this response and, say, leadership frodo, is that the triggering event is the engaging the enemy, not the readying. There is also no "then" to imply an order of events (i.e. engage enemy, ready, then apply boost).

It also helps then to engage more than one enemy with higher threat through card effect, as he can ready once and get a +2 or more boost.