Issue with having 9 teams is one team gets a free pass. 8, 16, 32, etc is the best for tournaments as no one gets a free pass.
you could do 3 groups of 3 teams.
The teams who win their group join a group to fight for the 1st-3rd place.
The teams who finish 2nd of their group join a group to fight for the 4th-6th place.
The teams who finish 3rd of their group join a group to fight for the 7th-9th place.
If you do for example 3 groups A,B,C, group for the 1st place = Winner Group A,B,C and same thing for the other groups. So each team will play 4 matches, In your format, They would have played 3 matches.