Half of the fast-food chain's nearly 900 restaurants remain closed. Many of those still open are offering limited menus or have cut their hours.
A spokesperson for the company said: "We anticipate the number of closures will reduce today and over the coming days as our teams work flat out all hours to clear the backlog.
"Each day more deliveries are being made; however, we expect the disruption to some restaurants to continue over the remainder of the week, meaning some will be closed and others operating with a reduced menu or shortened hours."
Earlier, the firm admitted it is "too early to say" how long it would take to reopen all of its branches.
Asked when customers could expect service to return to normal, a spokesperson said: "Our teams are working flat out all hours to get the rest back up and running as soon as possible – but it’s too early to say how long it will take to clear the backlog.”
KFC first apologised for the lack of the chain's key ingredient on Saturday, blaming "teething problems" with its supplier, DHL.
Fiona Cincotta, a senior market analyst at City Index, said the switch was "a case of cost cutting gone too far."
"Obviously, DHL were able to provide a more attractive price for distributing the chicken, but at what cost?" she said. "This will prove to be a costly mistake for KFC as customers are rarely forgiving towards these cost cutting errors.
"Furthermore, the majority of restaurants are run as franchises, owners will need to dig deep in order to weather the storm also begging the question as to how to deal with staff. Whilst this is unlikely to have a big knock on effect to the surrounding economy, it will most certainly put more than a few noses out of joint."
KFC said staff on short-term contacts working in restaurants owned by the chain would be paid the average hours worked per day over the past 12 weeks, while those on salaries would be paid as normal.
However, 80 per cent of KFC outlets are run on a franchise basis.
"Franchisees will be seeking their own independent advice, but we're encouraging them to adopt this policy too," the chain said.
The closures were the result of delayed chicken deliveries by DHL, which announced in November it had been appointed alongside QSL to manage the supply and distribution of food products and packaging for more than 850 KFC restaurants throughout the UK.
"Due to operational issues a number of deliveries in recent days have been incomplete or delayed," DHL said in a statement.
"We are working with KFC and our partners to rectify the situation as a priority and apologise for any inconvenience this may have caused."
KFC was previously a client of Bidvest Logistics.
The chicken chain detailed the problems in an earlier statement, which said: "We've brought a new delivery partner on board, but they've had a couple of teething problems - getting fresh chicken out to 900 restaurants across the country is pretty complex!"
"We won't compromise on quality, so no deliveries has meant some of our restaurants are closed and others are operating a limited menu, or shortened hours," the company added.
"We know that this might have inconvenienced some of you over the last few days, and disappointed you when you wanted your fried chicken fix - we're really sorry about that."
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