A method includes storing data in a group of analog memory cells by writing first storage values to the cells. After storing the data, second storage values are read from the cells using one or more first read thresholds. Third storage values that potentially cause cross-coupling interference in the second storage values are identified, and the third storage values are processed, to identify a subset of the second storage values as severely-interfered values. Fourth storage values are selectively re-read from the cells holding the severely-interfered values using one or more second read thresholds, different from the first read thresholds. The cross-coupling interference in the severely-interfered storage values is canceled using the re-read fourth storage values. The second storage values, including the severely-interfered values in which the cross-coupling interference has been canceled, are processed so as to reconstruct the data stored in the cell group.