I have found the following cyclic solution for your volleyball tournament which has the with and against balance that you are looking for. I suspect other schedules may exists for 6n+1 players, i.e. 19, 25, 31 players etc.
Court 1 Court 2 Bye
( 2 11 4 v 13 5 10) ( 6 7 8 v 12 3 9) ( 1)
( 3 12 5 v 1 6 11) ( 7 8 9 v 13 4 10) ( 2)
( 4 13 6 v 2 7 12) ( 8 9 10 v 1 5 11) ( 3)
( 5 1 7 v 3 8 13) ( 9 10 11 v 2 6 12) ( 4)
( 6 2 8 v 4 9 1) (10 11 12 v 3 7 13) ( 5)
( 7 3 9 v 5 10 2) (11 12 13 v 4 8 1) ( 6)
( 8 4 10 v 6 11 3) (12 13 1 v 5 9 2) ( 7)
( 9 5 11 v 7 12 4) (13 1 2 v 6 10 3) ( 8)
(10 6 12 v 8 13 5) ( 1 2 3 v 7 11 4) ( 9)
(11 7 13 v 9 1 6) ( 2 3 4 v 8 12 5) (10)
(12 8 1 v 10 2 7) ( 3 4 5 v 9 13 6) (11)
(13 9 2 v 11 3 8) ( 4 5 6 v 10 1 7) (12)
( 1 10 3 v 12 4 9) ( 5 6 7 v 11 2 8) (13)
For the benefit of others, I think the 'close' schedule you refer to is the one for 12 players found here in reply #7.