If a coworker mentioned a death in their family and you verbally gave your condolences, would it be "overkill" to give them a card?

Definitely send them a card, and in a month or so, once the novelty has worn off.... you know they're still hurting.
Follow up with them. Ask them how they're handling it. It will mean the world to them.
Remember we are relational beings. our deepest need that can be met by other humans is to be seen, acknowledged, and loved.

Some of the most meaningful interactions are from strangers who took a risk. remember that.