The wire is most likely stapled. That's the first problem.

I presume since the house was built in the 60's you are using
BX cabling???? If you are planning on using romex then
the outlet boxes that you are connecting to will not be
to code. They are specifically made for armored cable.
Unless you plan on using armored again then you'll have issues.

Getting the old outlets out is a real pain in the *ss because they are
nailed into the studs. What you can do is remove them and put
in remodel plasic boxes using romex. Granted its not stapled inside
the walls but such is life. \

Tom