The fast food chain said it was "working flat out" to reopen stores after there were issues with deliveries on Friday and over the weekend.
Its problems appeared to have deepened on Monday as an update on its website listed just over 80 UK restaurants, out of a total of 900, that were open at midday.
However, that number had grown to more than 300 by Monday evening.
On Sunday night 270 branches were open.
A spokesman said it was "too early to say" how long it would take to reopen those that had been closed.
The chain also said that where possible, staff would be "redeployed onto alternative tasks or to other restaurants that remain open".
It added: "We're encouraging team members to take holiday where possible but not forcing - the decision is up to them."
Salaried employees will be paid as normal, while others on short term contracts will be paid the average hours worked per day over the last 12 weeks, the company said.
It said it was encouraging franchisees to adopt the same policies despite operating their own contracts of employment.
KFC last week switched its delivery contract from Bidvest to DHL.
In a light-hearted statement on its website informing consumers about the problems, it said: "The chicken crossed the road, just not to our restaurants."
It added: "We've brought a new delivery partner onboard, 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.
"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."
It added: "The colonel is working on it."
DHL said: "Due to operational issues a number of deliveries in recent days have been incomplete or delayed."
The company said it was working "to rectify the situation as a priority and apologise for any inconvenience this may have caused".