We demonstrate temperature-controlled spectral tunability of a partially-pumped single-wavelength random laser in a solid-state random laser based on DCM [4-dicyanomethylene-2-methyl-6-(p-dimethylaminostyryl)-4H-pyran] doped PMMA (polymethyl methacrylate) dye. By carefully shaping the spatial profile of the pump, we first achieve a low-threshold, single-mode random lasing with an excellent side lobe rejection. Notably, we show how temperature-induced changes in the refractive index of the PMMA-DCM layer result in a blue shift of this single lasing mode. We demonstrate spectral tunability over an 8nm-wide bandwidth.