Summary: Recently, both the bilinear decompositions \(h^1(\mathbb{R}^n)\times\mathrm{bmo}(\mathbb{R}^n)\subset L^1(\mathbb{R}^n)+ h_\ast^{\Phi}(\mathbb{R}^n)\) and \(h^1(\mathbb{R}^n)\times\mathrm{bmo}(\mathbb{R}^n)\subset L^1(\mathbb{R}^n)+h^{\log}(\mathbb{R}^n)\) were established. In this paper, the authors prove in some sense that the former is sharp, while the latter is not. To this end, the authors first introduce the local Orlicz-slice Hardy space which contains \(h_\ast^{\Phi}(\mathbb{R}^n)\), a variant of the local Orlicz Hardy space, introduced by A. Bonami and J. Feuto [in: Recent developments in real and harmonic analysis. In honor of Carlos Segovia. Boston, MA: Birkhäuser. 57–71 (2010; Zbl 1202.46032)] as a special case, and obtain its dual space by establishing its characterizations via atoms, finite atoms, and various maximal functions, which are new even for \(h_\ast^{\Phi}(\mathbb{R}^n)\). The relationship \(h_\ast^{\Phi}(\mathbb{R}^n)\subsetneq h^{\log}(\mathbb{R}^n)\) is also clarified.