Many people are lust for that Hammond Growl, a combination of Hammond, Leslie and distortion.
If you want just a little, the first step is to adjust the preamp output level of your organ. (Also, of course, you can adjust the level control on your Leslie(s).)
Some people have experimented with replacing various tubes in the preamp (and/or Leslie amp) with other higher gain tubes. I have not tried this. Reports from those who have have not been compelling.
KonZissis has experimented with using a passive zener-diode passed circuit. See Kons12AX7basedEQ&OverdriveCircuit and KonsZenerDiodeBasedOverdriveCircuit.
BradBaker has gotten good results by 
changing the output tubes
in the Leslie amplifier from the stock 6550s to EL34s.
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