I started out by writing a post called "when are the Saints going to trade Ingram", but after thinking about it there is no way they will trade Ingram. The main reason is the Saints RB's are very talented but injury prone. Sure RB is a position that has injury written all over it, but there is no RB on the roster that has not dealt with an Injury. Ingram being the youngest and not really injury prone would be a good guy to keep. Pierre Thomas and Chris Ivory have had injuries and Sproles does not have the size to pound the ball in short yardage situations. There are 4 RB's on the roster and we have seen that when you have that many there is no way all of them can get into a rhythm, which apparently is needed for a RB to perform well. Andrew has shown in his post that the yards per carry were not that bad this season, and obviously if the defense thinks you are throwing, running is easy. But in my opinion that is the wrong way to go about it.
There are 3 options for trade. Sproles is too versatile and a favorite target by Brees, so he stays, no question.
PT: I do not see him being traded, however he only has two years left on his contract. They will possibly have difficulty keeping him on this low salary he has now in the future. Trading him is an option as the Saints have cap problems next year again when they need to re-sign Graham.
Ingram: As said, he is the guy you can count on to stay healthy. As Bush has shown us, when you run a Heismen trophy winning RB a LOT more he will show you stuff you have not seen before. So run him more! (Too many RBs on the roster??? Case in point).
Ivory: He is the most likely to be traded. He is a good runner. The RB market is bad, but still you want to trade him. Find a good backup with 1 or 2 years of experience that is under contract for 2 more years and trade Ivory for a player that adds depth.
I just realized there is stil this guy named Cadet. I was about to write you need a 4th guy on special teams (where you can use speed) and have him there in case a guy goes down. Cadet is a lesser version of Sproles. He sure has a future and you do not wanna get rid of him, as he is also very cheap. He stays on the roster and you put him in whenever a guy goes down... and there will be a guy going down.
So, the only thing to do is to get Ivory officially under contact (preferably for 2 years). Also have him sign for less than the tender in this year so you free some cap space for as long as he is under contract.
But as to answer the question in the title of this post: after the draft. The Saints will see which team has a surplus on guys they lack and they will know what position they want to add some experience in depth.