Join Date: Jul 2005
Location: Boulder County, CO
Bikes: '79 Gios, '80 Masi, '06 Felt, early '60s Frejus
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If you can't get the chain to the outermost cog, the first thing you want to do is loosen the cable, all the way. Even better, disconnect it. While you're down there, loosen all the barrel adjusters on that cable (turn clockwise until they stop).
Now, if the chain still won't reach the outermost cog, loosen the outer limit screw ("H") until it does.
After satisfying yourself that the chain does run in the outermost cog, ratchet your rear shifter until it's in its loosest position, the highest, smallest outermost cog. You want all tension out of the cable. Now re-fasten the cable at the derailleur while pulling as much slack as possible by hand. If you use pliers or any other mechanical means to pull slack, you'll probably over-tighten, so just pull with your fingers.
Now you're ready to work on the shifting. First, pull remaining visible slack out of the cable by tightening the barrel adjuster (anti-clockwise). Next, try to shift to the next cog. If it doesn't go in, shift back down and tighten the barrel adjuster about a quarter-turn (anti-clockwise). If for some reason it over-shifts (shifts two cogs), shift back down and loosen a quarter turn. Repeat the process of shifting up, then shifting back, and turning the barrel adjuster in increasingly small increments until you can get the chain running quietly in the second cog.
Keep repeating fine adjustment of the cable through all the cogs, using the barrel adjuster. When you get to the last (innermost) cog, adjust the inner ("L") limit screw so you can easily and consistently get the chain onto that cog without over-shifting.
Viola, you're done. If this isn't working you probably have a broken shifter or bent derailleur hanger.