Intermediate Approach is not quite the same as the Genkis. Its target audience includes both people who have and have not completed the Genkis. As a result, many grammar points from the Genkis are reintroduced at different points throughout the text, along with, of course, other more difficult stuff. While the Genki approach feels a lot more like a textbook, the Integrated approach feels like you're getting tossed into a real world of Japanese. There are tons of news articles, more conversations, etc, and far fewer practice exercises. You read stuff, it gives you vocab and explains grammar, but you have to try to use the material in real life to cement it.
As far as I know, there is no sequel. But, at some point, having a textbook becomes redundant. When you finish Intermediate appraoch, you should be able to learn grammar via dictionaries and look up vocab via a J-J dictionary (maybe with help from J-E-J). Your reading practice will be real sources(manga, newspapers, stories, etc) and speaking practice real conversations
(note: I havent finished intermediate approach yet. I got sidetracked by one of the JET program language books on passive/causative, and spend too much time reading other things. I'll pick it up again and finish maybe it August?)
So yeah, good luck.