Lattice-QCD simulations and theoretical arguments hint at the existence of an intermediate phase of strongly interacting matter between a confined hadron gas and a deconfined Quark-Gluon Plasma (QGP). We qualitatively and semi-quantitatively explore and differentiate the phase structures in the temperature window from the QCD pseudo-critical temperature to the pure-gluonic deconfinement temperature . We propose a three-regime picture using a hadron resonance gas (HRG) description augmented with the glueball spectrum based on the analysis of a large number, , of colors. We estimate the entropy density from our model to confirm that the lattice-QCD data are bracketed with three regimes, i.e., a hadron gas, a QGP, and a new phase for . In this new phase that we name a Spaghetti of Quarks with Glueballs (SQGB), thermal degrees of freedom of quarks are deconfined, yet gluons remain confined in glueballs. Since the Hagedorn temperature, , is universal in the meson and the glueball sectors, in the infinite limit, the phase diagram in the plane of the baryon chemical potential and the temperature is reduced to the one with the confined and deconfined phases and Quarkyonic Matter at high density. At large but finite , an SQGB window may open between these phases. We point out that the SQGB has interesting similarities with Quarkyonic Matter and that this matter in the large limit is confined as measured by the interaction between heavy quarks, but behaves in other respects like a quasi-free gas of quarks. As a result of the extrapolation to , we present a revised phase diagram with the SQGB phase bounded by thermal crossovers. Finally, we give a quantitative analysis of chiral symmetry restoration in the SQGB phase.