Incense and Peppermints by the Strawberry Alarm Clock, which scored about three months after "Light My Fire," sounds like it might have been inspired by the Doors and Manzarek.
I’d say there were contemporary bands on the Sunset Strip in the 1960s that were more in league with the keyboard sound of the Doors than Strawberry Alarm Clock:
The Music Machine- Come On In
http://www.youtube.com/watch?v=A_g84ubkwUg
The Music Machine- Eagle Never Hunts The Fly
http://www.youtube.com/watch?v=UKCl8cN90Kk
The Seeds - Up In Her Room
http://www.youtube.com/watch?v=eOWCmRuD8HY
Sky Saxon Blues Band (formerly the Seeds) - The Moth And The Flame
http://www.youtube.com/watch?v=hQ8SzrzRc4k
Sky Saxon Blues Band - I’ll Help You (Carry Your Money To The Bank)
http://www.youtube.com/watch?v=hQ8SzrzRc4k
The lead singers of each passed away in recent years (Sky Saxon of the Seeds dying the same day as Michael Jackson and Farrah Fawcett).