your idea to build up the concrete base around the bottom of the post would probably work to stablize the post. However, I think you need to make sure the two layers of concrete are connected to each other by some means other than the rotting metal post. You should probably drill some holes into the concrete base and insert vertical lengths of rebar (perhaps even using an epoxy to set them firmly into the old concrete) and extend them up into the new pour. This would give some horizontal stability to the two surfaces.
Even though the posts are metal and not wood, Steve is likely still correct about the cause of the rot. Make sure the concrete you pour around the posts this is slanted away at the top so the water doesn't say against the post's surface, or even find a path down between the post and the concrete. Good luck!