In case you are not aware, it is possible to play 11 rounds without repeats.
With 20 rounds, it is possible to have each player playing with 25 other players exactly twice, and opposing the remaining 10 players exactly once. There is an example schedule below.
(23 10 4 11) (35 13 9 26) (28 7 5 12) (24 14 36 21) (2 18 33 3) (16 20 15 29) (22 27 30 32) (17 8 25 19) (6 34 1 31)
(24 6 5 12) (31 14 10 27) (29 8 1 13) (25 15 36 22) (3 19 34 4) (17 16 11 30) (23 28 26 33) (18 9 21 20) (7 35 2 32)
(25 7 1 13) (32 15 6 28) (30 9 2 14) (21 11 36 23) (4 20 35 5) (18 17 12 26) (24 29 27 34) (19 10 22 16) (8 31 3 33)
(21 8 2 14) (33 11 7 29) (26 10 3 15) (22 12 36 24) (5 16 31 1) (19 18 13 27) (25 30 28 35) (20 6 23 17) (9 32 4 34)
(22 9 3 15) (34 12 8 30) (27 6 4 11) (23 13 36 25) (1 17 32 2) (20 19 14 28) (21 26 29 31) (16 7 24 18) (10 33 5 35)
(17 34 22 7) (23 5 32 31) (30 19 36 33) (24 35 21 3) (18 15 4 28) (25 8 6 14) (27 1 11 26) (10 12 9 29) (2 20 13 16)
(18 35 23 8) (24 1 33 32) (26 20 36 34) (25 31 22 4) (19 11 5 29) (21 9 7 15) (28 2 12 27) (6 13 10 30) (3 16 14 17)
(19 31 24 9) (25 2 34 33) (27 16 36 35) (21 32 23 5) (20 12 1 30) (22 10 8 11) (29 3 13 28) (7 14 6 26) (4 17 15 18)
(20 32 25 10) (21 3 35 34) (28 17 36 31) (22 33 24 1) (16 13 2 26) (23 6 9 12) (30 4 14 29) (8 15 7 27) (5 18 11 19)
(16 33 21 6) (22 4 31 35) (29 18 36 32) (23 34 25 2) (17 14 3 27) (24 7 10 13) (26 5 15 30) (9 11 8 28) (1 19 12 20)
(19 22 2 26) (13 20 4 14) (25 24 3 11) (6 5 30 18) (35 17 9 32) (8 12 33 15) (1 27 10 21) (23 34 29 16) (31 7 36 28)
(20 23 3 27) (14 16 5 15) (21 25 4 12) (7 1 26 19) (31 18 10 33) (9 13 34 11) (2 28 6 22) (24 35 30 17) (32 8 36 29)
(16 24 4 28) (15 17 1 11) (22 21 5 13) (8 2 27 20) (32 19 6 34) (10 14 35 12) (3 29 7 23) (25 31 26 18) (33 9 36 30)
(17 25 5 29) (11 18 2 12) (23 22 1 14) (9 3 28 16) (33 20 7 35) (6 15 31 13) (4 30 8 24) (21 32 27 19) (34 10 36 26)
(18 21 1 30) (12 19 3 13) (24 23 2 15) (10 4 29 17) (34 16 8 31) (7 11 32 14) (5 26 9 25) (22 33 28 20) (35 6 36 27)
(22 18 13 23) (2 29 35 1) (27 16 12 25) (11 30 28 32) (10 36 5 3) (19 7 4 21) (6 24 26 8) (34 33 15 14) (31 20 9 17)
(23 19 14 24) (3 30 31 2) (28 17 13 21) (12 26 29 33) (6 36 1 4) (20 8 5 22) (7 25 27 9) (35 34 11 15) (32 16 10 18)
(24 20 15 25) (4 26 32 3) (29 18 14 22) (13 27 30 34) (7 36 2 5) (16 9 1 23) (8 21 28 10) (31 35 12 11) (33 17 6 19)
(25 16 11 21) (5 27 33 4) (30 19 15 23) (14 28 26 35) (8 36 3 1) (17 10 2 24) (9 22 29 6) (32 31 13 12) (34 18 7 20)
(21 17 12 22) (1 28 34 5) (26 20 11 24) (15 29 27 31) (9 36 4 2) (18 6 3 25) (10 23 30 7) (33 32 14 13) (35 19 8 16)