Dielectric loss spectra at both ambient and elevated pressure are presented for propylene glycol (PG), dipropylene glycol, and tripropylene glycol. While in previous literature the excess wing (EW) was observed only for PG, we show herein that under high pressure an EW (coexistent with a secondary beta relaxation) is present for all these materials. Furthermore, at higher pressures the EW evolves into a well-separated peak. The pressure sensitivity of the EW is very different from that of the beta relaxation, being close to that of the structural relaxation, whereas the beta peak is virtually insensitive to pressure. These differences imply a very different degree of intermolecularity of the underlying motions. A similarly small sensitivity to pressure has been observed for the beta relaxation in sorbitol. We have measured the spectrum of sorbitol at high pressure, and likewise it indicates the presence of a submerged EW. Finally, the behavior observed for the EW in the glycols is shown to be consistent with predictions of the coupling model, relating the most probably relaxation time for the secondary process to a noncooperative relaxation time. This suggests that the EW may be the precursor to the alpha relaxation and therefore of fundamental importance to understanding the vitrification process in supercooled liquids and polymers.