As soon as Mr Irrelevant's name is called and the draft ends (probably before), teams start burning up the phones trying to recruit all the undrafted guys that they still have interest in. It's more or less a free for all where teams put in offers and do their best to recruit the guy to give their camp a shot.
The Philadelphia Eagles began sending letters to borderline draftable players about a year ago, laying the foundation for signing them after the draft.
The letters detailed Philly's interest in a player, and their intention to contact said player after the draft if he is not selected.
For a fringe prospect, you'd probably rather go undrafted than get taken in the 7th round, because at least you have some control over your situation and can pick a team that might give you the best shot to make a roster. The Eagles in particular have had some decent luck with UDFAs over the years, most notably with Quintin Mikell. So, really anything to get a little edge in the process is smart.