Can you make a metal denser by melting it and make it cool down under high pressure? The answer is probably no. But can you make a metal denser by melting it and make it cool down under high pressure? Or would it just crack under internal stress when it returns to a solid? I'm going to assume that the metal doesn't have air bubbles or other impurities inside it and that we're talking about charted and known densities.
 A: First of all, high pressure (i.e., compressive equitriaxial stress) is simply not going to induce cracks (or failure of any sort) in a uniform solid—there's just nowhere for the material to move to reduce the sum of strain energy and surface energy.
Yes, melting a metal and then freezing it might increase the density in several ways, although the improvement might be negligible relative to simply applying the high pressure. Here are a couple of examples:
First, one might eliminate voids in the material. For example, one can substantially increase the density of a metal foam by melting it and refreezing it as a solid body by applying pressure.
Second, if the material has been thoroughly cold worked, i.e., it contains a large number of dislocations and grain boundaries, then these defects would contribute some slight excess volume that would be removed by melting and refreezing. However, this free volume is estimated to be quite small (think 0.1%) even in cases of severe plastic deformation. The resulting maximum achievable increase in density would be comparable.
Third, as @By_Symmetry notes below, the high temperature might promote a pressure-induced transformation to a higher-density alternative crystalline phase that might have been kinetically limited at lower temperatures.
Note, however, that if you heated the metal and then cooled it down very quickly (i.e., you quenched it), you might actually decrease the density relative to the case of no thermal processing, either by kinetically trapping vacancies (which are much more prevalent at high temperatures), by producing an amorphous structure (by kinetically limiting the material's tendency to freeze into a crystal), or by locking in a relatively low-density alternative crystalline phase.
