Proof-theoretic method is one of the primary sources for constructive proofs of interpolation. After first being used to prove Craig interpolation using cut-free sequent calculi, the method has been extended to Lyndon interpolation, and uniform interpolation, as well as to analytic sequent calculi. More recently, it has been extended to hypersequents, nested sequents, labelled sequents, yielding, for instance, a positive answer to the open problem of whether Gödel logic (intermediate logic of linear frames) has Lyndon interpolation. An interesting feature of this extension is its essential use of Kripke semantics (for modal or intermediate logics). While the algorithm for contructing interpolants is purely proof-theoretic, the proof of its correctness relies on the semantics. More pertinently, the algorithm in general and individual interpolant transformations in particular are guided and shaped by the structure of Kripke models of the logic in question. In this work, we extend this syntactico-semantic method to proving uniform interpolation for K, D, and T using nested sequents and for S5 using hypersequents.