A 6 way split would be a drop of 10.5 db (the ones I have seen are made by a 2-way to a 3-way) so adding another 2 way splitter to one of the lines may be too much signal drop for some of the digital channels. Once you get it hooked up double check the new line and the split line and go through all the channels look for blocking or stuttering, both on the higher and lower channels. It's been a few years but when I worked with Mediacom we only provided 20db of signal to a home to begin with so by the time you have drop losses and line losses and your 10db of of splits today then the signal might just be too weak.
Could work perfectly too, especially if the tap is running a bit hot.
6 ways will lose 9db. Internal 2way (3.5 db loss) to 2 balanced 3 ways (5.5db loss). You could just get a 8 way splitter to replace the 6 way. 1.5 more of db loss really shouldn't affect you unless you are real close on the signals.