With the information you provide, the answer has to be a bit generic.
Lets call A the site for non logged in and B the logged in site.
The first thing to consider, is position of objects, for instance if you have a left or right menu on B, but there is no need for in on A, then just leave that space white on A. It it's too wide, you can reduce a bit the space since the perception won't be affected and the use of A would be improved.
If you can provide generic information on A, the use it as place holder for the equivalent interaction on B. On A it would just be an image, and on B you provide a way to interact with that image/graph/block.
If there are navigation links that are going to appear on B, as a block, then try to leave that space empty on A. If you can't, make obvious that those links are "new" on B. By that I mean, make them different from the common links. For instance, you can put them on a box on the left or right side, or if they are on the horizontal menu, put them on the right or left, all together.
If there are too many, just create a new bar that goes below the common one and make it visibly different.
If each page is going to have a lot of changes, then there is no way to do it smoothly.