Picking to trays should be a little easier and connected more into the system.

Step 1. Make using trays to pick more time efficent.
An alternative way that would be more efficient would be to have a button that says "Pick to Tray". Picker loads all orders, after scanning the 1st item they are requested to "Scan a Tray". They can scan any tray they want 1 time. This will link that order to that tray and register the item being placed in that tray. The next item for that same shipping number will display the tray the order need to be placed in since it was already assigned.

Step 2.
At the packing station allow the shipper to scan the tray to call up the order vs the shipping number on the pick ticket.

This system will allow picking jobs to be assigned to pickers without the need to have paper copies of the orders. Allow the picker to pre sort all of the orders while picking. Allow the shipper to scan the tote and be able to ship the order. This would give the possibility to pick and ship orders paperless and would save time. Additonally you already have most of the funcationality built into the system, just need to access it in a different way.