An MTZ file is the electron density map for the given molecule. This can sometimes be obtained from the electron density server. If the map can't be obtained simply load any MTZ file for any molecule.
Find the carbon atom on the residue you wish to continue the backbone from.
This is not the alpha carbon, it's the one connected to the oxygen
Click on the add residue tool then on the carbon atom
Coot attempts to add a new residue that fits the electron density map. This initial placement is usually junk because the reason the author didn't include the residue in the first place is because the electron density here is too spread out. Also, refinement steps may have removed the electron density here completely.
Move the residue close enough to bond to the carbon atom
Click on the regularize zone tool then on the new residue
This idealizes the residue's geometry while ignoring the electron density map
Choose File → Save Coordinates to save
Duplicating a portion of a molecule onto another molecule
Suppose you have two PDB files of the same protein in different conformations but one is missing the last few residues. You want to insert the residues and have this section of the protein match exactly between the two PDB files.
In PyMol load the protein with the residues to be copied.
Make a new PDB file that is just these residues and the residue they connect to.
Load this PDB file and the file that is to receive the residues into Coot.
In Coot choose Calculate → LSQ superpose and align the molecule and the fragment using the common residue between the two
Delete the common residue from the fragment
Choose Calculate → Merge Molecules to merge the two into one molecule
Choose File → Save Coordinates to save this new molecule