What I tend to do actually helps quite a bit, especially as a support main.
First of all, you need to keep playing solo queue (sorry).
Second of all, when you get a team that either a) wins overwhelmingly or b) has a player or two you mesh extremely well with (or also do absolutely great), ask the team if they want to play again.
Thirdly, keep playing with the team as much as you can, switching out people as either better people come along, the current teammates start to rage/be toxic, or the teammate just leaves.
It is kind of like speed dating, but for teams.
Also, it is easier to establish these types of teammates (and also to keep them and help them play their best) if you try to be positive, forgiving, and also assist in map awareness. My map awareness, for example, is quite excellent... unless a skirmish suddenly happens bot lane, in which case I divert all my attention to that. I can always use any help I can get.
The only thing I will recommend to be cautious about though is to not proceed with this until your own skill at the given role/champion is at a point which greatly exceeds the rank you should be in. For example, my Soraka in my elo is absolutely disgusting. 5.09 KDA with a 63% win rate, but regardless, the ability to keep track of everyone (i.e. who to heal, positioning, chaining my silence field to snare, etc.) and cooldowns in teamfights is what could completely turn around games. It was only when I reached that height of skill in my elo that I began to employ this tactic.
Good luck. I hope this helps. Cause it did help me. ^.^