My orchid seems to have sprouted another stem, as well as lots of roots. I know I'm going to have to re-pot but is there anything I can do to encourage the new stem to flower?
It looks like your Orchid is just finishing flowering on the one stem. Generally when the Orchid is done blooming it will need a rest or dormant period.
You can trim or prune away the spent flower stem and wait to see if the new stem produces any flowers.
Here are some links with more information.