A mechanism for controlling a measurement threshold used for triggering handover of a user equipment device (UE) when the UE is connected with a first access node, the first access node operating on a first frequency band and a second frequency band, and a second access operating on the second frequency band. A determination is made as to whether the UE supports inter-band dual-connectivity service with the UE being connected with the first access node on the first band and with the second access node on the second band. And responsive to the determination being that the UE does not support the inter-band dual-connectivity service, the measurement threshold is adjusted from a default level to an adjusted level to help facilitate handover of the UE from being connected with the first access node on the first band to being connected with the first access node instead on the second band.