The present study discusses effects of two parameters: freestream Reynolds number and trip height on transitional hypersonic boundary layers due to diamond-shaped trips at Mach 6. This work is an extension of the baseline flow and trip configurations. To accurately capture flow physics, a high-order, low-dissipation scheme for the convection terms in the Navier-Stokes equations is used. No forcing apart from the trip is applied to perturb the boundary layer in the present study. Analysis of power spectral density (PSD) confirms that the source of instability downstream of the trip is the coupled system of the counter-rotating streamwise vortices and the shear layers originating from the top and sides, and the wake of the trip in all the parametric cases. Therefore, the source of instability is independent of freestream Reynolds numbers and trip heights considered in the current study. However, a higher freestream Reynolds number case leads to higher peak-amplitude (PSD) frequencies, and correspondingly, higher perturbations are observed at a fixed downstream location. A range of peak-amplitude PSD frequencies associated with instability at different trip heights is observed around the peak-amplitude frequency of the baseline configuration. This finding is similar to isolated cylindrical trips at near-critical roughness heights. Furthermore, the mean recirculation region upstream of the trip and the mean instability-onset location downstream scale linearly with trip height in the current trip configuration. However, the mean transition-onset location varies nonlinearly with trip height.