Babergh and Mid Suffolk councils have restarted some waste collections for some properties on major roads on Wednesday - but there are still problems on minor roads and residential streets across the districts.
Officials are urging people to leave out bins if they are due to be collected and if the teams are able to pick up the rubbish they will be emptied.
Ipswich council is continuing to collect waste where they can - about 65% of properties due to be collected had their bins emptied on Tuesday and its teams will be trying to collect those missed over the rest of the week.
Bin collections in East Suffolk are still suspended because of the difficulty in reaching properties on minor roads. The situation in the district is being reviewed every day.
The storm did not affect West Suffolk so badly - and that council's teams have been able to continue working.
Comments: Our rules
We want our comments to be a lively and valuable part of our community - a place where readers can debate and engage with the most important local issues. The ability to comment on our stories is a privilege, not a right, however, and that privilege may be withdrawn if it is abused or misused.
Please report any comments that break our rules.
Read the rules here