One of these should do the trick. The first three columns would be in the 1st half of each round.
9 or 10 players
Rd: Pairings:
1: 1-10 6-5 9-2 7-4 8-3
2: 10-6 2-1 5-7 3-9 4-8
3: 2-10 7-6 1-3 8-5 9-4
4: 10-7 3-2 6-8 4-1 5-9
5: 3-10 8-7 2-4 9-6 1-5
6: 10-8 4-3 7-9 5-2 6-1
7: 4-10 9-8 3-5 1-7 2-6
8: 10-9 5-4 8-1 6-3 7-2
9: 5-10 1-9 4-6 2-8 3-7
11 or 12 players
Rd: Pairings:
1: 1-12 7-6 11-2 8-5 10-3 9-4
2: 12-7 2-1 6-8 3-11 5-9 4-10
3: 2-12 8-7 1-3 9-6 11-4 10-5
4: 12-8 3-2 7-9 4-1 6-10 5-11
5: 3-12 9-8 2-4 10-7 1-5 11-6
6: 12-9 4-3 8-10 5-2 7-11 6-1
7: 4-12 10-9 3-5 11-8 2-6 1-7
8: 12-10 5-4 9-11 6-3 8-1 7-2
9: 5-12 11-10 4-6 1-9 3-7 2-8
10: 12-11 6-5 10-1 7-4 9-2 8-3
11: 6-12 1-11 5-7 2-10 4-8 3-9