Last year, there were 3 trades on July 1, 2 trades on July 2, 1 trade on July 9, and then no more trades in the entire NHL for the rest of the month of July. And only 5 of those trades involved at least one NHLer.
There were 4 trades in August 2023, one of which was an AHL trade.
There were 2 trades in September 2023, one of which was an AHL trade.
There were 3 trades in October 2023, two of which were AHL trades.
Trades can happen after the first week of July. But, the volume drops way off when compared to the week before the draft through July 7 or so.
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I took 2020 out due to the league calendar being really messed up. But, here are the trades league-wide that involved NHLers over the past 5 "normal" offseasons.