You will be much, much happier with the final result if you can re-render the tiff sequence with an alpha channel. You really don't want to use AE as a Band-Aid if you can do it properly the first time.
Aside from what Dave said, you are using a pretty awful workflow. If the BG is white, then simply using a luma key, a lumam matte, Shift Channels, Channel Mixer and a million other effects would already provide a much better result and create genuine transparency/ Alpah plus allow unpremultiplying/ decontaminating the fringes as well...
If you're stuck creating a matte this way (and I agree with the others that you should look at alternatives), then using the Refine Matte effect can help you to improve the edges of your matte through smoothing and feathering.