This shows a misunderstanding of how a tree grows a ring. Whatever problem you had was not related to the time of year. When the snow is just melting the outer ring is the same as it was at the end of the previous growing season.
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No, this wasn't the problem. Upon inspection it was obvious that I was either using early wood on the back or a paper thin latewood ring. I might be off by a few days or weeks on the timing, as it was 6 or 8 years ago this happened, but I'm sure this is what was happening. All the other growth rings were fat, solid, and well established. These showed a grainy layer covered by a very thin layer.
We do get drought years and the like around here. None of the trees I had cut from this same grove gave me any issues and they were all cut in the winter, summer, or fall.