Provided are a computer program product, system, and method for allocating additional requested storage space for a data set in a first managed space in a second managed space. A request for additional storage space is received for a requested data set stored in a first managed space in the storage. A revised amount of storage space for the requested data set comprises at least an amount of space currently allocated to the requested data set in the first managed space and the requested additional storage space. If the revised amount of storage space exceeds a value, then allocation is made of the revised amount of storage space in allocated storage space in a second managed space of the storage. The data set is stored in the allocated storage space in the second managed space.