Health workers are to stage a second national strike on 26 September - during the Labour Party conference.
NHS Logistics supplies medical equipment and food to the NHS
The protest, by workers from the NHS Logistics agency, will follow a 24-hour walk out this Thursday - the first national NHS strike for 18 years.
It is feared the protests, over the sell-off of the network to German firm DHL, could lead to English hospitals running short of vital equipment.
But the government said it believed any impact would be limited.
Further industrial action will follow.
Around 900 staff were balloted by Unison and of the 66% who voted, 74% backed a strike.
NHS Logistics supplies around 50,000 lines of products, including bedpans, latex gloves, syringes, bandages, medical equipment and food.
Hospitals have to order equipment three days ahead and have limited space to store stockpiles.
Sense of loyalty
Karen Jennings, Unison's Head of Health, said: "The NHS will be a hot topic at this year's Labour Party Conference and Unison's contemporary motion highlights the outsourcing of NHS Logistics to DHL.
"Taking strike action on the very day that health is to be debated by the Labour Conference will put our members' protest centre-stage.
"They have worked hard to build NHS Logistics into an award-winning service and don't deserve to be treated in this way. There is simply no logical explanation for this transfer."
The government believes the DHL deal, which comes into force at the start of October, will help save £1bn over 10 years.
But Unison views it as a sign of "creeping NHS privatisation".
The union said it expected any action to hit the supply of goods to hospitals, but added it had agreed emergency plans to cover life-saving equipment.
The Department of Health said most hospitals also had their own supply and delivery arrangements and would be able to "cope with demand".
A spokesman said: "The NHS used around 500,000 different products such as catering supplies, office equipment and medical supplies but only around 51,000 of these products are provided by NHS Logistics.
"The majority of hospitals have their own local supply and delivery arrangements.
"The impact of a one-day strike is not expected to cause significant disruption to NHS operations.
"The impact will be similar to coping with a bank holiday.
"However, we have put detailed contingency plans in place to ensure minimum disruption to NHS services in the event of industrial action."
Workers who will walk out are based at distribution centres in Alfreton, Derbyshire; Runcorn, Cheshire; Normanton, West Yorkshire; Maidstone, Kent; and Bury St Edmunds in Suffolk.