Hexagonal two-dimensional materials with broken inversion symmetry (as BN or transition metal dichalcogenides) are known to sustain chiral phonons with finite angular momentum, adding a further useful degree of freedom to the extraordinary entangled (electrical, optical, magnetic, and mechanical) properties of these compounds. However, because of lattice symmetry constraints, such chiral modes are constrained to the corners of the Brillouin zone, allowing little freedom for manipulating the chiral features. In this paper, we show how the application of uniaxial strain leads to the existence of unique chiral modes in the vicinity of the zone center. We also show that such strain-induced chiral modes, unlike the ones pinned at the K points, can be efficiently manipulated by modifying the strain itself, which determines the position of these modes in the Brillouin zone. The present paper results add a technique for the engineering of the quantum properties of two-dimensional lattices.