There are many more. Sticking with the Reprise era for the time-being (since it's fresh in mind):
R20-157 California / America The Beautiful
First issued as a white label promo with picture sleeve in 1963. Then as a commercially labelled 45 in 1966, of which just 1000 were made for Governor Brown's reelection campaign in California.
0493 Gunga Din / Gunga Din
White label promo of Sinatra reciting passages from Gunga Din (1966). Just 300 were pressed and given away to friends.
CN-101 Ring-A-Ding Ding / Nothing But The Best
From Cal-Neva Lodge (1962-63). Orange and white sleeve with Sinatra's name printed above an outline of the states of California and Nevada. This was a souvenir given away to guests at Sinatra's resort during the tourist season.
PROS-1 35mm Demonstration Record
Thick paper sleeve with cover promoting new 35mm recording process and displaying four new release covers, one of which is Sinatra Conducts Music From Pictures And Plays
. The disc inside contains one selection from each album. It is a 33-1/3 rpm recording.
PROS-2115 Stereo Review Salutes Frank Sinatra—1984
Black, white, and gray promo sleeve contains 33-1/3 record with four Sinatra selections. It was offered for $5 through mail order only in an ad by the magazine and 2000 were made.
PRO-674 Night And Day / Everybody Ought To Be In Love
12-inch disco mix, 45 rpm.
U-21863M It's Time For You
Purple label (no name) promo for Chrysler commercial (ca. 1980).
The latter song is discussed in the Frank's Recordings thread for Promotional Issues and Samplers. There is bound to be some overlap between that thread and this one.
kb-2077/2078 "High Hopes" with Jack Kennedy / Jack Kennedy "All The Way" (Jimmy Joyce Singers)
Both tunes with special new lyrics for the 1960 presidential election campaign. Black (non-picture) sleeve; green (no name) label. (No FS
on the B side.) Recorded at Capitol: see discussion below.
Note: This is by no means an exhaustive list of vinyl promos; I only tried to pick out some interesting examples.