Franco was not interested in an immediate monarchic restoration. His goal was to create a new state. He proclaimed that his rebellion was in the name of the fatherland, not the monarchy. In a letter to Alfonso XIII, Franco described a potential monarchic restoration as "a goal of which we have a premonition, but one which we cannot yet see because it is so far away" (un camino cuya meta presentimos pero que por lo lejana no vislumbramos todavía).

Here is a brief article in Spanish

See also Bowen's Spain During World War II, pp. 73-75.

When Franco won the war he fused the Falange (Fascists) and the Carlistas Monarchists* into one political party under his control. The other groups that supported the rebels (Monarchists, CEDA, nationalist party, etc) were ignored and prohibited, this way Franco got all the power as a dictator under the new fascist party.

Of course many monarchists complained but the cruel repression silenced them. By the 60/70s the monarchism was a weak movement and nobody thought that restoration of monarchy was possible anymore.

*The Carlist Monarchists did not support the official line of monarchy of the moment (Alfonso his son Juan), but an old claim to throne of a separete line by Carlos, and it was mainly a Basque movement.