Interesting idea. The limit would be that if you use the new vent fan to simply blow the exhaust into the under-tub space (which exhausts through the floor under the tub) it might flow backward through the existing fan and back into the bathroom. To prevent that, you could also turn on the existing vent fan, and have both fans running.
The only disadvantage to this is that the existing fan is noisy. You could, of course, replace the existing fan with a quieter fan. Computer fans are cheap.
Or, I think you are suggesting that you might add the new fan, direct it through a new hose, and direct the end of the new hose through the floor under the tub. That would solve the back-feed problem. But to do it, you would need to remove the wood panel between the toilet and tub. I would hesitate to do that, since space in that area is tight.
Bill
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