So, there are a couple possibilities. The first is that you have reached the limits of the amp, and the second is that you've reached the limits of the headphones. If it is the second, then there is no solution to the problem other than changing to different headphones. If it is the first, it depends on whether you are hearing distortion due to the amp running out of voltage, or current. The KGSSHV uses +/-450 volt or +/-500 volt power supplies, so its voltage limits are higher than the BHSE (which uses +/-400 volt PS), and close to or the same as the DIY T2 - in fact, there is less than 1 dB difference between the max voltage output of the KGSSHV with 450 volt PS and the DIY T2. The alternative is that you are hearing distortion due to current limiting, which could be alleviated by increasing the standing current in the KGSSHV.
However, a more important point is, that if you are listening regularly at "deafening volumes", you are likely irreparably damaging your hearing. One sign of this is that if you hear ringing in your ears after a listening session, that is, as someone colorfully put it, "the scream of a dying acoustic nerve." You can replace headphones and amplifiers. You cannot replace your ears. Once your hearing is gone, it is gone.