I think they absolutely should.
At his age, and dip in production, he's no longer worth his $6.5 million salary...but at a new deal for a year or two at $2.5-3 million per year, I think that is reasonable for a veteran scoring presence that has always had a knack for finding the next gear in the playoffs.
Even with his dip in production and age, I still think he has the tools to be a 50 point player, and his acquisition would not cost the Jackets any assets. He would add depth to the forward group, and may make some players currently on the roster more expendable as movable trading pieces, or buyout (Umberger) allowing the Jackets to move up in the draft or acquire other assets.
The downside to this acquisition would be obviously his age and regressing offensive statistics, but he is still a capable forward, and would be worth a shot as a free agent. The biggest upside is that he doesn't cost the team an asset to acquire, he would be another veteran voice in the room alongside Vinny Prospal.