if you go straight to ice water you will have a bigger initial temp differential and will drive temp down faster. the only problem is you need more ice. IMO this is easier than switching cooling sources partway. just load the sump up with ice and be prepared to add some if you need too.
One needs lots more ice. I found that 30+ pounds of ice were needed to do that. Fortunately had access to a freezer where ice could be made and didn't have to resort to store-bought at $2 per 10 pound bag. Simply filled a 5 gallon bucket a couple quarts at a time until it was nearly full and frozen. If you try to freeze the entire bucket at one time the ice's expansion will tear it apart. It took days to fill the bucket.
Switching over is a PITA, but my homebrewer's inherent cheapness cringes at spending so much on ice. I figure 10 minutes from boiling to 110 with straight tap water and another 20 with ice to get it down to the mid 60's isn't too shabby.
Also experimented with frozen PET bottles as the sole source of cooling for the IC. It takes a lot of them and also much more time than just cooling with tap water. Might be an option in water scarce areas, but IMO this approach is a homebrew dead-end.
I had to use 3 times the amount of bottles (nearly a 5cu freezer full) in the photo below to cool 12 gallons of wort from boiling to pitching temps. Took almost 2 hours with constant supervision. I could have achieved the same results in 30 minutes with tap water and a switch over to around 15 pounds of ice. I figure it takes about a pound per gallon if one switches over to a recirc after reaching 100F. YMMV