Spare parts to pick up.
1. Extra cylinder hand (pawl) with spring.
2. Extra bolt/trigger spring. (The one shaped like a tuning fork).
(Both the above springs can also be made/bent, out of piano wire if necessary).
3. An extra trigger guard screw, because on the 1858 Remy, there is only one trigger guard screw to remove to get to the bolt/trigger spring, and this is the screw you will remove and reinstall the most often, so nice to have a spare in case the slot on yours gets boogered.
4. As already mentioned, a nipple wrench.
(Grease the nipple threads with a little (very sparingly) lithium grease or anti seize grease, teflon spray applied with your finger works too, so they can easily be wrenched out for complete cleaning).
5. Paperclips, diaper pins and sewing needles are handy for cleaning the nipples' large and small hole.
6. Wouldn't hurt to pick up an extra new hammer with the cam on it. Eventually your hammer cam will wear out, and it is much easier to just put in a new hammer that has a new cam on it already, than trying to drive out and press in a new cam that has to be pressed in, in the exact proper angle in the hammer. Been there with that situation on my 1860s'.
You can get most of the spare parts you will need, as a kit, as I believe one poster has already mentioned.
That's about all I can think of.
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