Future measurements, motivated by several open physics questions, will be greatly enhanced by the upgraded performance of the iTPC. Searching for a possible critical point in the QCD phase diagram is one of the major scientific tasks in heavy ion physics. The critical point, if it exists and can be identified, would provide a landmark in the phase diagram and guide further experimental and theoretical studies of QCD matter under a wide range of conditions. The ability of RHIC to scan a wide range of beam energies and beam species provides a unique opportunity for the exploration of the phase diagram. RHIC has completed Phase-I of the beam energy scan program (BES-I) with center-of-mass beam energies of 39, 27, 19.6, 14.5, 11.5 and 7.7 GeV. The STAR BES Phase-II (BES-II) White Paper sets out the physics case for an in-depth study of energies below 20 GeV with typically 20 times the statistics as in the same energy region in BES-I. Thus the enhanced measurement capabilities of STAR after the iTPC upgrade are a vital part of the new BES-II effort. The iTPC upgrade extends the rapidity coverage by 50%, a major benefit for many analyses, especially fluctuations (Kurtosis) and baryon v1 measurements; it improves the 2nd-order event-plane resolution away from mid-rapidity by a factor of 2, greatly enhancing all elliptic flow measurements; and in the area of dielectron measurements it reduces hadron contamination from a dominant source of uncertainty (20%) down by an order of magnitude, much less than the expected statistical uncertainty (10%).