Its sounds more like a solo organist, (There were many Hammond organists around that era) with backing added in the studio. (To me it seems very much in the style of Klaus Wanderlich)
Try posting on one of these forums, http://organforum.com/forums/default.aspx http://www.organfax.co.uk/community/ as they will probably be better able to recognise the player and tune.

Regards

Bill