@needrain - have you determined that what is coming from the flower stem is another bloom?
Sometimes they throw keikis rather than a new bloom. A keiki is a new baby orchid. Them or more blooms can be encouraged using a special paste, though at times - like you case - the Phal decides to do so on its own.
Right now of my 14 Phals, 3 have keikis. 2 are pushing a year old and I will likely soon remove them and "plant" them on their own.
Welcome to the world of growing orchids. From what I have read in this thread, you should do ok when you move it indoors. Any questions we will try to help. In case of failure, it happens to all of us.
cheers,
david
