Personally, if one end is that wet, I would get some pallets, tear apart one or two to make the tops of the others solid, and put them in the wet corner/end so that the bedding doesn't get saturated. You want it to compost (so some moisture), but not to mold!
So, I would rake/hoe out - at least the wet end completely, and the dry end some (maybe 1/2, as you do want some 'starter' microbes for the deep litter method), put the pallets (with solid tops by adding extra boards in the spaces) down on the wet end, then top up all the bedding. You might have to trim a pallet so that they fit together inside the run based on the width.
New ramial chips are good - if they are really coarse, then a light coating of shavings on top.
Just my thoughts. Do what seems reasonable and practical to you - as what I am suggesting - on top of the roof now - is a lot of extra work. That said, long term it will probably make it easier on you and more comfortable for the chickens - with less extra shaving needed - in the future. I am constantly playing 'clean-up' in my uncovered run with all the rain we are having - and it is a LOT of work, and with the ducks and their wet evacuations, it just gets disgusting if I don't between them and the constant rain - especially in the area that I can't just willy-nilly add bedding due to the gate.
