I looked at your photo a little closer. If it were me. I think I would mow the rye in order to release the clover. Don't mow shorter than 6" or so and don't mow if the plot is experiencing stress from heat or drought.I agree with everything you are saying. My neighbor had cattle in this area in August and September so I did not need to mow. I applied glyphosate. About a week later I lightly tilled up the thatch. I then broadcast the cereal rye seed and fertilizer. I did have a soil test and applied fertilizer accordingly. I broadcast a small 5 lb. bag of Imperial Whitetail Clover no till mix around the perimeter and got the exact result you mentioned. The clover mix didn't do much last fall but came on this spring. It is now buried in that tall rye.
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How much sun does that plot get?
My best looking clover right now is the stuff that is not exposed to all day sun. A little shade is a good thing during mid summer.