If you want the skin of your puppet to fit the foam structure, you will have to have sew different pieces of fabric together.
When you do this, it creates seams. These seams can be hidden, however, using a stitch Jim Henson developed.
This stitch is often referred to as the Henson Stitch, Puppet Stitch, or Ladder Stitch as seen in this link by TheCreatureWorks.
This stitch works best on fabrics that are fuzzy. If you try to use this on non fuzzy fabric you will get a strange zigzag effect on the seam. Fuzzy fabric hides this.
This stitch can be used while the fabric is on or off the foam base.
For this demonstration by Tom Stewart, a thread color different than the color of the fleece is used so you can see what is being done.
1. Place the two pieces of fabric side by side, wright sides up.
2. Put a knot in the end of your thread that has been waxed using bee's wax so your thread won't tangle.
3. Push your needle through the fabric starting from the under side. Pull the thread all the way to the knot. This should be about 1/8 of an inch or less from the edge of the fabric. (if the knot comes out, make the knot bigger and try again.)
4. Take the needle and push it into the other piece of fabric from the top. Do not push the needle all the way through.
5. Catch a small amount of material on the needle, like a quilter would do, and pull the thread through. Do not pull this stitch tight.
6. Now go back across to the other piece and, starting close to the first stitch, repeat the above step.
7. Do this for 5 to 6 stitches; keeping the stitches loose.
8. Pull the stitches tight. This pulls the fabric together.
9. With the tip of the needle, gently pull out the fleece that was trapped in the seam. Be careful not to pull out your stitches.
10. Repeat this process all along the seam.
You now have a seam that is very hard to see and gives your puppet a professional look.