You can probably replace the felt, yourself, by spray-gluing it on.... using head liner spray glue from your local auto parts store. Do a test gluing on some scrap wood to see how this glue applies (sprays) and how the fabric lays down (start in the middle and work your way toward the edges. Spray the substrate (wood), allow to dry about a minute or so, then press the felt onto the wood. Again, do a test gluing to get a feel for the nuances of the application. When testing, check closely that the glue doesn't imprint itself through the felt... in other words, that the "wetness" of the glue doesn't show on the outer surface of the felt. With some thin fabrics, that spray glue will show (resembling a wet spot, if you can understand what I mean) through the fabric, especially if you use too much glue.
For this type of application. contact cement can be used, also. Might be difficult to apply the cement to the felt, though.
DON'T stretch your fabric, but do apply some slight tension as you lay it out. Felt generally doesn't stretch, very much, but just be careful not to stretch it to the point of deforming the layout.
For a more specific technique, as to this spray-gluing, check with your local pool table repair store/outlet. They might give you some application tips or demonstrations... or a local auto repair guy might give you some (similar) tips about head liner installation. Otherwise, an upholsterer may help/assist you.
Not sure what grade of felt you'll reinstall, but your local pool table guys might have a better grade of appropriate felt, than your local fabric store. Doesn't hurt to check with several outlets, though, for decent felt.