I think... maybe she's not so much trying to hide the card from Ennis as from herself. Does that make any sense? What I mean is, maybe she doesn't have any real intention of hiding the card from Ennis, but that doesn't mean she wants to be staring at it, or that she's willing to just hand it to him. Maybe she's being a wee bit passive aggressive, putting it where he could overlook it.
I think there is a lot of great subtle acting of that nature in the role of Alma. This reminds me of a discussion I read one time as to why Alma picks up her purse, as if she's going somewhere, right after she sees Jack and Ennis kissing. It doesn't make so much sense, logically. She's not going anywhere, and she knows she isn't. Ennis has already vetoed her 'knife and fork' idea, and they don't have a sitter, and she wouldn't have walked out and left the girls in any case. But reaching for her purse... I think it's an automatic, instinctive, and partly symbolic thing. She wants to flee the situation, even though she can't. So she picks up her purse in preparation, as if she's got somewhere to go.
I think it's the same with the postcard. She's not really taking action. It's just a small, private gesture, symbolic of what she'd really like to do - to make Jack, and his influence on both their lives, conveniently disappear. It's like she's thinking with her actions.
It also shows the powerlessness of Alma. She can only make vain little gestures toward what she'd really like to do..