Basically, no. The calendar reflects the current career date, and only increments as a result of upgrading components whilst in base. The actual patrol start date is not calculated (or perhaps more correctly not applied) until you load the patrol.
Although, there are scenarios where the date *will* reflect the next patrol start date, like where you have upgraded components that have extended your days in base beyond the default 28, 31 or whatever SH3Cmdr has (randomly) set.
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