Quantitative speed-of-sound (SoS) and attenuation of tissues are closely related to pathology; however, conventional B-mode images are limited to qualitative visualization. Existing ultrasound full-waveform inversion (FWI) methods for quantitative SoS reconstruction are primarily developed under double-sided or ring-shaped arrays, which limits their applicability to widely adopted routine clinical acquisitions. In this work, we develop a frequency-domain, total variation (TV)-regularized FWI framework tailored for single-sided linear ultrasound arrays, which enables quantitative reconstruction of SoS maps using standard clinical probes. To address the severe ill-posedness and computational challenges in this setup, efficient forward modeling, fast gradient evaluation, ADMM-based optimization, and multi-GPU parallelization are integrated into the inversion framework. Numerical experiments in a thyroid cyst imaging scenario demonstrate that the proposed method reconstructs the SoS of both simple (fluid-filled) and solid cysts with improved visual and quantitative performance compared to conventional FWI. Additional 2D and 3D simulations across different target and array apertures further elucidate the capabilities and limitations of single-sided ultrasound FWI.