This is a sectional beam, so I have wound all the new warp on the warp beam. I am tying on at the back of the loom to the old warp. The old warp goes from the front of the loom, through the reed, through the heddles and to the back of the loom. I do have a loose knot through the old warp to keep it from accidentally sliding through the reed. Once I have all my knots tied, I will gently pull the new warp through the heddles and reed to the front. I have inserted eye heddles, so the knots go through fairly easily, but I do take care and just pull a 1 or 2 inch section through at a time. Once I have the new warp at the front of the loom, I cut off the old warp and lash on. Hope this helps! I don't know that this saves a lot of time, but the threading will be correct. Also, at my advanced age, I find it more comfortable to tie knot rather than thread heddles and reed -- also my eyesight isn't great.
Yes, I am warping from back to front here. I have wound on the new warp onto my sectional beam, and I am then I tie the new warp onto the old warp at the back of the loom. Then I pull it through the heddles and then the reed to the front beam. You can also do this if you warp from front to back, but of course you wouldn't have a sectional beam for that.