I've dug thousands of plugs in the turf. The most efficient, cleanest, turf saving way for my types of ground I typically hunt at is to make a hinged door plug.
I use a lesche digger and cut a semi-circle plug (about 180 degrees, or slightly more depending on the type of turf), then lift the cut side up (like a hinged door). To keep the plug from dropping back in the hole, I use a long probe (screw driver will work) to pin it. I then place all the dirt I'm going to take out of the hole onto a piece of rubber truck tire inner tube that I cut into a square(a towel will work well also, but I prefer the water repellent properties of the rubber). Now, depending on the type of soil and its moisture content, I may or may not be able to get all the dirt I removed back into the plug in order to make my plugged (hinged) grass remain as level as possible with the original turf, which is a very important step. The reason it's important is that if it's sticking up a little bit, it has a chance of being sucked up into the commercial lawnmowers (common at most parks I hunt), leaving a very unsightly dirt hole in the turf. So typically ,I'll place 90-95% of the dirt back in the plug before tamping it down with my foot to make it flush. The remaining dirt I simply spread out next to a nearby tree, or slowly release it (like watering) out of my rubber mat as I continue to hunt until it is all gone. I remember just a few weeks ago, I saw a really dry, turf-less, dry spot (about 18" x 12") in this one meadow I was detecting....it was an eye sore...not sure how it got there, but I kept dumping my extra plug dirt in this spot as I hunted around it......within one week, on my return trip back to this meadow, new, young grass was growing right from that dried up area now that there was fresh dirt there to retain moisture and create fertilization. Made me feel great!!
It may sound like a lot of extra work, but it's really not. My tools are like an extension of my arm.....you get very efficient with the plugging process....been doing it now for over 25 years. I've made thousands of 5-9" plugs in the turf, and trying to keep the turf looking like it was untouched is my highest priority. I've seen/witnessed far too many plugs where it looked like the highest priority of the digger was the treasure in the hole, ignoring any awareness/concern of the turf.
Good Luck!!
CAPTN SE
Dan