Collections across the borough were suspended from Wednesday due to staff absences.
A number of the waste crews were either self-isolating or poorly with Covid-19 at the time the decision was taken.
The service was completely suspended until Monday next week, but the council now says more waste crews have tested positive and there will need to be further changes next week.
A spokesperson said: “Household waste (green bin) collections which did not take place this week (w/c Jan 25) will now be taking place next week. There will be no recycling (blue bin) collections next week for any area of the borough.
“However, those residents with a purple paper and card bin will be collected as normal.
“We ask all residents who had their green bin missed this week to put it out for collection by 5.30am on Monday, February 1.”
Because the council cannot give a precise date for the collection, they ask that the bins are left on the kerbside until they are collected.
Overflow of household waste can be presented bagged at the side of your bin, the council says, asking residents to “please keep checking your bin and take it in as soon as it has been emptied”.
Normal collections are due to resume the following Monday - February 8 - in line with the collection calendar.
The spokesperson went on: “We are aware that the majority of the recycling (blue) bins will be full by your next scheduled collection so we will accept additional bags of recycling presented with your blue bin. Once again, we apologise for any inconvenience this may cause.”
Assistant Director for Operations at the Council, Victoria Burgess, said: "We would like to thank everyone for their ongoing support to our crews and we are working hard to solve the issues we are facing around accumulation of household waste. I would ask residents to help our crews next week by not presenting bins that don’t require emptying. This will enable us to collect from properties more quickly.
“Please could you share this message with friends and family in the Borough as well as any elderly/vulnerable neighbours to make sure they have also picked up on this new announcement.
“Our focus in doing so is helping residents to manage their residual waste the best they can; whilst giving our team the time they need to recover from Covid-19.”
If you have waste or recycling that you cannot keep until your next recycling collection, Lincolnshire County Council has agreed to extend the opening days at the Household Recycling Centre on Bittern Way for the disposal of black-bagged general waste and recycling only.
The two extra days of opening are Wednesday and Thursday of each week up until Thursday, February 18, between the hours of 8am and 4pm. People are asked not take any garden waste.
To access one of these slots please book at www.lincolnshire.gov.uk/clickandtip
Keep an eye on local media, the Council’s website – www.bostonuk.com - and social media channels for updates.