Okay Redcloud,
I will move away from the "hire a professional" and give you some basic advice. I don't know your specific skill level, your specific building codes nor the specific steps pertaining to your specific project so I will give you some things to think about. I just wanted to come back and post this so you don't think I am some high falutting contractor that does not understand the DIY concept because I do, just so you know.
You may be able to lift the areas in question with some lally columns and shore up the floor joists and support them with sufficient support required by your municipality and by code. I do not know why the inspector is involved in the project but he is the one that you will have to consult with on the w's pertaining to the project and they probably will tell you that they are not contractors and you need to consult with one but on the other hand he might just give you some pointers. They tend to shy away from giving ANY advice because of the liability especially with a project like this.
This is a project that a homeowner can do but the risk in my opinion is not worth it. You may also have to have certain credentials to work on basements in your municipality so I just want you to know that there are so many factors at play here it is hard to tell you which way to go.
I understand that this is a DIY site and that you have come here for help and I understand that I am offering advice in your situation to not DIY however I hope that you can see why I would do so in this situation. Your house probably isn't going to collapse however you can do damage to the house if you do not know what you are doing. Plus the expense associated with this kind of[project can get rather high although you can probably rent Lallies or use beams and hydraulic jacks to tackle this project but with a project like this if you are at a DIY site asking, "how do I do this?" then my opinion is that you should hire it done and I realize that money does not grow on trees however if the inspector is involved then there is something going on that we are unaware of.
Good luck and please be safe.