In Paper I, we presented an overview of the Southern-sky MWA Rapid Two-metre (SMART) survey, including the survey design and search pipeline. While the combination of MWA’s large field-of-view and the voltage capture system brings a survey speed of ∼450 deg2 hr−1, the progression of the survey relies on the availability of compact configuration of the Phase II array. Over the past few years, by taking advantage of multiple windows of opportunity when the compact configuration was available, we have advanced the survey to 75% of the planned sky coverage. To date, about 10% of the data collected thus far have been processed for a first-pass search, where 10 min of observation is processed for dispersion measures out to 250pc cm−3, to realise a shallow survey that is largely sensitive to long-period pulsars. The ongoing analysis has led to two new pulsar discoveries, as well as an independent discovery and a rediscovery of a previously incorrectly characterised pulsar, all from ∼3% of the data for which candidate scrutiny is completed. In this sequel to Paper I, we describe the strategies for further detailed follow-up including improved sky localisation and convergence to timing solution, and illustrate them using example pulsar discoveries. The processing has also led to re-detection of 120 pulsars in the SMART observing band, bringing the total number of pulsars detected to date with the MWA to 180, and these are used to assess the search sensitivity of current processing pipelines. The planned second-pass (deep survey) processing is expected to yield a three-fold increase in sensitivity for long-period pulsars, and a substantial improvement to millisecond pulsars by adopting optimal de-dispersion plans. The SMART survey will complement the highly successful Parkes High Time Resolution Universe survey at 1.2–1.5 GHz, and inform future large survey efforts such as those planned with the low-frequency Square Kilometre Array (SKA-Low).