You may be unable to find your gf in the Social Security Death Index
because your gf died too soon.
First, social security as enacted in 1935 covered only
employees. In the 1950s and 1960s, that was extended to
self-employed persons, and finally to just about everyone. At about
the same time, one's social security number became one's Tax
Identifying Number, so an additional number of people who did not
have social security numbers were forced to get one in the 1960s. So
it is possible that your gf did not have a social security number if
beginning 1936 he was an independent businessperson or worked for a
government entity or a church, then retired in the 1950s before
social security numbers became more general.
Second, SSDI does not really get going until the early 1960s. Some
previous deaths are recorded, but it does not appear to be
comprehensive at the time your gf died. My hunch is that the number
of deaths listed >from the 1950s is significantly less than those for
the 1960s, and that the last half of that decade is more
comprehensive than the first half.
You probably want your gf's social security number to seek your gf's
application for a social security number, which should list his city
of birth and his mother's maiden name. By the late 1950s, some
states had spaces on their death certificate forms for social
security numbers, so you might check your gf's death certificate for
San Diego CA