The spurt how behavior of a linear high-density polyethylene was carefully studied in capillary experiments. Depending on the flow conditions, new types of complex transients were observed. The study pointed out a second oscillating area, described by a second hysteresis cycle, close to or inside the classical one. In one case, the second oscillating area was situated between two parts of the stable second branch, and pressure oscillation patterns for the two distinct oscillating areas were clearly reported. In another case, the smaller secondary oscillations occur within the cycle of the primary oscillations, showing double instability pressure patterns. The transients in the capillary flow of linear polyethylene put in evidence a complexity, which cannot be explained only in terms of melt compressibility associated with a simple relationship between slip and shear stress.