The Herefordshire strawberry growing co-op came under the spotlight earlier in the year after the TGWU received complaints from 200 migrant workers about employment conditions.
Retailers launched an urgent probe and gave S&A a clean bill of health on the major allegations.
But the TGWU stepped up its campaign last week, picketing stores in cities in England and Scotland.
S&A had not taken the steps necessary to avoid a repeat of the issues during next year's strawberry season, said deputy general secretary Jack Dromey. "We have no confidence that exploitative practices by S&A Produce have been eradicated for good."
The union said it had amassed 3,000 signatures after a first wave of picketing and vowed to keep the campaign going until there was progress in discussions with S&A and its customers. There is no question of taking S&A produce off the shelves, a Tesco spokeswoman said. But the store did identify some minor problems at S&A during an unannounced spot check earlier in the week. "We have agreed a plan with S&A to support them in putting things right. We will continue to check standards with regular independent spot checks."
Sainsbury's said it would keep sourcing from S&A, but was committed to talks with the T&G over a solution to the stand-off.
S&A rejected the latest allegations by the unions as unprofessional and dishonest, claiming it was simply fishing for new members.
HR director Gilbert Savory said the wrangle was taking its toll on the company, which was not supplying any produce at present. "There have not been costs from losing customers, but in the time lost dealing with this."