Synthetic transmit aperture beamforming is an increasingly used method to improve resolution in biomedical ultrasound imaging. Synthetic aperture sequential beamforming (SASB) is an implementation of this concept which features a relatively low computation complexity. Moreover, it can be implemented in a dual-stage architecture, where the first stage only applies simple single receive-focused delay-and-sum (srDAS) operations, while the second, more complex stage is performed either locally or remotely using more powerful processing. However, like traditional DAS-based beamforming methods, SASB is susceptible to inaccurate speed-of-sound (SOS) information. In this paper, we show how SOS estimation can be implemented using the srDAS beamformed image, and integrated into the dual-stage implementation of SASB, in an effort to obtain high resolution images with relatively low-cost hardware. Our approach builds on an existing per-channel radio frequency data-based direct estimation method, and applies an iterative refinement of the estimate. We use this estimate for SOS compensation, without the need to repeat the first stage beamforming. The proposed and previous methods are tested on both simulation and experimental studies. The accuracy of our SOS estimation method is on average 0.38% in simulation studies and 0.55% in phantom experiments, when the underlying SOS in the media is within the range 1450-1620 m/s. Using the estimated SOS, the beamforming lateral resolution of SASB is improved on average 52.6% in simulation studies and 50.0% in phantom experiments.