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We present the inverse scattering transform as a tool to solve the initial-value problem for the focusing nonlinear Schr\"odinger equation with one-sided non-zero boundary value $q_{r}(t)\equiv A_{r}\, e^{-2iA_r^2t+i\theta_r}$, $A_r\ge 0$, $0\leq\theta_{r}<2\pi$, as $x\rightarrow +\infty$. The direct problem is shown to be well-defined for solutions $q(x,t)$ to the focusing nonlinear Schr\"odinger equation such that $[q(x,t)-q_{r}(t)\vartheta(x)]\in L^{1,1}(R)$ [$\vartheta(x)$ denotes the Heaviside function] with respect to $x\in R$ for all $t\ge 0$, for which analyticity properties of eigenfunctions and scattering data are established. The inverse scattering problem is formulated both via (left and right) Marchenko integral equations and as a Riemann-Hilbert problem on a single sheet of the scattering variables $\lambda_{r}=\sqrt{k^2+A^2_{r}}$, where $k$ is the usual complex scattering parameter in the inverse scattering transform. Unlike the case of fully asymmetric boundary conditions \cite{DPVV2} and similarly to the same-amplitude case dealt with in \cite{GB13}, the direct and inverse problems are also formulated in terms of a suitable uniformization variable that maps the two-sheeted Riemann surface for $k$ into a single copy of the complex plane. The time evolution of the scattering coefficients is then derived, showing that, unlike the case of solutions with the same amplitude as $x\rightarrow \pm \infty$, here both reflection and transmission coefficients have a nontrivial (although explicit) time dependence. These results will be instrumental for the investigation of the long-time asymptotic behavior of physically relevant solutions to the focusing nonlinear Schr\"odinger equation with nontrivial boundary conditions, either via the nonlinear steepest descent method on the Riemann-Hilbert problem, or via matched asymptotic expansions on the Marchenko integral equations.