When there is a general trip of a multistage compression train when all ASVs' are auto opened by shutdown logic, I have only seen all ASVs' simultaneously open. But it would be ideal to sequence the opening as you suggest for general trip, maybe 1-2seconds lapse between each HP, IP and LP ASV. This approach is okay since for almost all cases of general trip, there is nothing blocking the path from one stage to the next.
It is not a good idea to wait for individual stage to go near surge before the stage ASV is open for general compression train trip.