Yes, they're not unlocked until Floor 2, and even then they're just mixed in with the others so you might not find any. Black Funguses drop Calm Bounty cards if you're not on Floor 1, they can be helpful.
Anyways, in the title you asked for a trick to get all the cards. Okay, this isn't easy, and you probably shouldn't worry about it until the end of the game. But the only way to get every card is to get every Enemy card, and there is a trick to that. It's just not a very good trick.
Enemy Cards are random, but the last enemy you kill is the enemy that will drop an Enemy Card. As a result, if you have an Enemy Card from a Heartless, kill that Heartless first so you don't get duplicates. Or at least, try not to kill them last. That's the best you can do! Getting all the cards is all about luck and being stubborn.
Enemy Cards are more likely to drop in Looming Darkness, Teeming Darkness, and Almighty Darkness rooms. Sora cannot get the Darkball card, so kill Darkballs first at all times. And don't worry about grinding for Enemy Cards at this point in the game, it's not worth it.