We ran into the same issue on the migration so our solution was do two
things.  Our backups start at 4pm.  I let them run until about 8pm and then
I lower the migration threshold.  At 10pm (backups finish around 10 or 11) I
kick off a script that makes a list of all of my disk volumes, divides it
into 4 scripts and then does move data's to my tape pool.  This gives me 5
sessions draining the disk pool and it generally finishes about 2am at which
point I kick in my backup to copy pools.  This kept us going to disk so we
can increase the number of threads yet let us drain in a timely fashion.