It all depends on where you plan to stop. See PMags' End to End Guide to the CT at http://www.pmags.com
At the end of segment 6, you can take the free bus right into Breckenridge. Re-supply locally at the grocery store and/or the outfitter.
At the end of segment 8, you can hitch into Leadville and do the same, or you can go on to Twin Lakes, where you're better off sending a re-supply box to the general store. You also go right by Princeton Hot Springs Resort near the end of segment 13, where you could send a box.
At the end of segment 14, Salida is reportedly one of the best town stops on the whole CDT/CT - full resupply possible locally.
Salida to Silverton is more complex. One option is to exit to Creede at San Luis Pass or to Lake City at Spring Creek Pass.
At Molas Pass, you can hitch into Silverton, which has an outfitter and grocery stores.