Does Sam trigger if you engage an enemy who only has a higher engagement cost after the +10 from Halbarad? Or does the engagement cost have to be lower when you select to engage them?

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.

It works.

Halbarad saved my Hobbits, against those enemies with low engagement cost. So Hobbit Cloak, and Staff of Lebethrone still works!

And because Sam's ability is a response, if you engane more ennemies, he gets boosted for each ennemy you engage (sorry about the confusing phrasing, English is not my native language...).