To add a serial number field to a transfer order line a user will have to dot walk from assets table. But doing this will not allow the user to see the serial number field until the transfer order line is submitted. Another option is that a user can add the serial number field to the reference list of alm_asset.
Adding Serial Number field to a Transfer Order Line
Steps to add the serial number to the reference list of alm_asset:
1) Got to sys_ui_list table
2) Filter for Table name: alm_asset and View: Reference List
3) Go to that record
4) Add serial_number to the related list 'List Elements' as shown in the attached screenshot