How Aegis would use swing-arm and vertical launchers together

Would a ship fire its swing-arm missiles first and use VLS only if that launcher became overwhelmed?

No. An Aegis command system would use the swing-arm and vertical launchers together.

The proposed firing sequence holds VLS in reserve until the swing-arm launcher becomes overwhelmed. In the mixed-battery example, an Aegis command system would employ both launcher types in the same engagement.