Hourly to Salary- paid incorrectly-best practice

Comments
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Assuming the employee is salary exempt starting 6/1. Simply changing to salary does not eliminate overtime if the employee is still non-exempt. Also, switching to salary does not necessarily eliminate the need to track hours or pay overtime.
My inclination would be to leave the original payment alone, note that the employee's status changed as of June 1, and issue a retro payment to pay the correct compensation. If the intent was to convert the employee to exempt, then noting that and paying the additional compensation should be sufficient. If the employee is non-exempt, the overtime should be recalculated to conform to the new regular rate of pay and the employee should receive the remainder of the salary and the additional amount of overtime compensation due based on the change in the regular rate of pay.
If the employee is paid under a fluctuating workweek arrangement, then additional overtime compensation is also due.