It is not really a new spike, they come from the base of the plant. Yes, cut the old spike off above your side branch but note that Phal. growers with lots of orchids cut the old spike off at the base to encourage the plant to grow a new spike.

Those side branches never produce the number of flowers as a new spike produced from the base.

More information in the link.

Here is a link that might be useful: FAQ on Phalaenopsis Blooming