Provided are a computer program product, system, and method for using an alias volume name for a volume to allocate space to a data set. An assignment of a plurality of volumes to a data set is maintained, wherein the volumes are configured in a storage system. A request is received to extend the data set. An alias volume name is assigned to the data set for a previously assigned volume to the data to extend the data set in response to one of the previously assigned volumes having available space to extend the data set. A base volume name is assigned to the data set for a volume not assigned to the data set in response to one of the previously assigned volumes to the data set not having available space to extend the data set.