The phrase "supplement the theory of" is correct and usable in written English.
You can use it when discussing how additional information or evidence can enhance or support an existing theory.
Example: "The new research findings aim to supplement the theory of cognitive development by providing empirical data."
Alternatives: "enhance the theory of" or "add to the theory of".
An evolutionary framework of vertical integration is needed to supplement the theory of the product life-cycle.
Another way of developing a more adequate account of impossible fictions is to stay classical but supplement the theory of Lewis 1979 with impossible worlds (classical accounts of impossible worlds include Nolan 1997b and Zalta 1997, among others).
This hypothesis would supplement the theory of this paper, in particular the occurrence of plaques at certain, rather specific locations such as bifurcations.
It was later supplemented by the theory of optimum utilization of production factors by the selection of production lines.
Using the nonlinear field theories of mechanics supplemented by the theory of finite growth, the latest brain models characterize the developing brain as a morphogenetically growing outer layer on a mechanically growing inner core (Budday et al. 2014).
Further works supplement the theory by extending the definition of connectivity from two nodes to k nodes [5].
Recall that propositional theories supplement this theory of reference with an extra layer with a theory which assigns a content, as well as a reference, to each meaningful expression.
The theory of the characteristic function of an optical system was further developed in three supplements.
The ASE-Model is based on the theory of Theory of Planned Behaviour (TPB) of Fishbein and Ajzen [19] and is supplemented by elements from the Social Cognitive Theory (SCT) of Bandura [20].
"The Theory of Everything"?
"M-theory is the theory of everything.
