There are schedules for
24 and 20 players here. But I am not sure if the 24 player schedule works for you as it uses 6 tables.
Also a schedule for
22 players here.
I have some options for 21 players below.
21 players, 9 rounds (all partnerships different, all opposition pairs different)
The 9 players who have a bye only play 8 games.
(21 7 v 10 6) ( 9 13 v 4 14) (19 12 v 16 15) ( 1 20 v 3 11) (18 17 v 5 2) ( 8)
(19 8 v 11 4) ( 7 14 v 5 15) (20 10 v 17 13) ( 2 21 v 1 12) (16 18 v 6 3) ( 9)
(20 9 v 12 5) ( 8 15 v 6 13) (21 11 v 18 14) ( 3 19 v 2 10) (17 16 v 4 1) ( 7)
( 9 11 v 2 6) (19 16 v 7 8) (14 10 v 4 12) (13 1 v 5 18) (21 15 v 3 20) (17)
( 7 12 v 3 4) (20 17 v 8 9) (15 11 v 5 10) (14 2 v 6 16) (19 13 v 1 21) (18)
( 8 10 v 1 5) (21 18 v 9 7) (13 12 v 6 11) (15 3 v 4 17) (20 14 v 2 19) (16)
(21 14 v 8 17) (18 6 v 4 20) (11 10 v 9 16) (19 5 v 12 3) (15 7 v 1 2) (13)
(19 15 v 9 18) (16 4 v 5 21) (12 11 v 7 17) (20 6 v 10 1) (13 8 v 2 3) (14)
(20 13 v 7 16) (17 5 v 6 19) (10 12 v 8 18) (21 4 v 11 2) (14 9 v 3 1) (15)
21 players, 12 rounds (all partnerships different, all opposition pairs at least once and at most twice)
The 12 players who have a bye only play 11 games.
( 4 1 v 5 8) ( 3 9 v 13 12) (17 6 v 16 11) (14 2 v 18 21) (19 20 v 7 10) (15)
( 5 2 v 6 9) ( 1 7 v 14 10) (18 4 v 17 12) (15 3 v 16 19) (20 21 v 8 11) (13)
( 6 3 v 4 7) ( 2 8 v 15 11) (16 5 v 18 10) (13 1 v 17 20) (21 19 v 9 12) (14)
( 7 13 v 8 18) (15 1 v 3 12) (21 11 v 10 16) ( 6 20 v 14 9) ( 4 5 v 19 2) (17)
( 8 14 v 9 16) (13 2 v 1 10) (19 12 v 11 17) ( 4 21 v 15 7) ( 5 6 v 20 3) (18)
( 9 15 v 7 17) (14 3 v 2 11) (20 10 v 12 18) ( 5 19 v 13 8) ( 6 4 v 21 1) (16)
( 6 1 v 17 19) ( 9 4 v 10 11) (16 2 v 12 20) (13 5 v 21 15) ( 3 8 v 14 18) ( 7)
( 4 2 v 18 20) ( 7 5 v 11 12) (17 3 v 10 21) (14 6 v 19 13) ( 1 9 v 15 16) ( 8)
( 5 3 v 16 21) ( 8 6 v 12 10) (18 1 v 11 19) (15 4 v 20 14) ( 2 7 v 13 17) ( 9)
(10 4 v 13 9) ( 2 3 v 17 8) (16 20 v 7 19) (11 18 v 15 6) (12 1 v 21 14) ( 5)
(11 5 v 14 7) ( 3 1 v 18 9) (17 21 v 8 20) (12 16 v 13 4) (10 2 v 19 15) ( 6)
(12 6 v 15 8) ( 1 2 v 16 7) (18 19 v 9 21) (10 17 v 14 5) (11 3 v 20 13) ( 4)
I do not have anything good for 23 players however. I can make something, but there might be as many as 10 pairs of players who never meet, either as partners or opponents. I consider this 'social balance' important, that is to say, the number of times people sit down together at the same table should be as balanced as possible, and all the schedules mentioned above have this property.
