A method for data storage includes, in a memory system that stores data by programming analog memory cells to respective programming states selected from a predefined set, providing a first read command that retrieves the data stored in target memory cells, and a second read command that identifies at least one potentially-interfering memory cell that is programmed to a programming state in a predefined subset of programming states expected to cause interference. Given data is stored in a first group of the memory cells. After storing the given data, the given data is retrieved by reading the first group using the first read command, reading a second group of the memory cells using the second read command, and compensating for the interference caused to the first group by the memory cells in the second group that were identified by the second read command.