I found an injured bird

Here is a SA website:
http://www.faunarescue.org.au/birds.htm
It may pay to give them a call. They should be able to give you a drop off location for the bird, or they will come and collect it from your home.
Basically just keep a towel or sheet over the cage so it is dark so the bird doesn't get scared. Other than that it sounds like you are doing the right thing so far.

Are you sure it is a native pigeon? If it isn't then it will be PTS straight away.

Keep her somewhere dark, warm and quiet. Make sure there is nothing that can stress her out, because that's what will be most likely to kill her.

Are there any wild life sanctuaries in your area? If there are, call them up and they might be able to get a carer out to pick up the bird. The quicker you get it to them, the more likely it is for her to survive.

Edit: In response to Abbie, the sanctuary I work at will usually accept non-native birds. So an SA sanctuary might to?

Hm maybe. Where I live they don't rehabilitate non native birds. We rang up several wildlife organisations, vets, the RSPCA etc. and they said they only care for natives.
They also said they assess all injured birds, and if they think that they won't be able to be released back into the wild, or it will take too long for them to recover then they are PTS.

I once did a rehabilitation course, and i asked them about the common laughing turtle dove(not native)..and they said they rehabilitate them cos apparently the birds of prey eat em...

I think it's wrong that sanctuaries won't take non-native birds. What if they're migratory birds?

I called fauna rescue (thanks abbie) and the lady I spoke to said that she was already looking after a pigeon so she couldn't take her. She said to keep the room dark and quiet (I have her in the bathroom) and put a blanket over the cage. Which is what I've done. She said the first 24 hours are the most crucial, I can look after it over night but i will try to take her somewhere if she's still alive in the morning.
I called her digit. she's lovely

I was looking at some pics, she looks like a brown cuckoo

Most vets an after hours number. Try any local ones. A lot of vets either won't do birds or don't know much about them though. But that would be better than leaving it until the morning. If it's a cat attack you'll need to get it to the vets as soon as possible because it may die from infection

http://birdsinbackyards.net/

This site might help with an id also