Since adopting the practice of 2-bath fix (I use an alkaline fix too, but not TF-4) I have had no problems in bleaching and toning, which are really like magnifiers of processing deficiencies.
I fix for 30s in fix 1, then transfer the print to a holding bath. At the end of the printing session, I re-fix all prints in fix 2 for 30s. (This is film strength fixer.)
The purpose of the first fix bath is to dissolve undeveloped silver; the purpose of the second bath is to bring it out of the fibers and into the solution. The beauty of this is, since the print is only in the fixer for 30s intervals with a holding bath inbetween, the dissolved silver compounds don't have a chance to sink deep into the fiber base, where they are difficult to wash out.
This is followed by a 10-minute clearing bath with agitation (which some will argue is not even necessary with the alkaline fix) and a 30 minute wash (or however long it takes to get some food).
Yes, you will get many different opinions and methods. I started the 2-bath fixing after reading Rudman's Toning book, the chapter on archival processing, and have since had no problems whatsoever. Your fixer will last about four times as long, also.