A half dozen of some unusual 45s I own are featured belows
Time Has Come Today-Chambers Brothers This is not the version you know. 1966 version, similar but shorter, with some unusual instrumentation.
Cloudy Summer Afternoon-Bud & Travis I have never heard this on the radio, found it used in the 1960s. Laid back folk song,
Lovin' Spoonful could have done it.
Barry McGuire did do it.

Won a radio station contest in 1967 and received a copy of
Wonderful Wino-George Carlin
This was a huge hit instrumental in San Bernardino in 1969. didn't get a copy until about ten years later.
The Minotaur-Dick Hyman 7 1/2 minutes of synthesizer, and listed as side 2

Love
Get Away-Georgie Fame Pop Jazz Rock song
The Joker Went Wild-Brian Hyland One of the first songs I heard on the radio and really liked. I never realized until recently that it was a Bobby Russell song
