To fabricate robust metallic nanostructures with top-down patterning methods such as electron-beam lithography, an initial nanometer-scale layer of a second metal is deposited to promote adhesion of the metal of interest. However, how this nanoscale layer affects the mechanical properties of the nanostructure and how adhesion layer thickness controls the binding strength to the substrate are still open questions. Here we use ultrafast laser pulses to impulsively launch acoustic phonons in single gold nanodisks with variable titanium layer thicknesses, and observe an increase in phonon frequencies as a thicker adhesion layer facilitates stronger binding to the glass substrate. In addition to an all-optical interrogation of nanoscale mechanical properties, our results show that the adhesion layer can be used to controllably modify the acoustic phonon modes of a gold nanodisk. This direct coupling between optically excited plasmon modes and phonon modes can be exploited for a variety of emerging optomechanical applications.