But first - Do not calculate new average and control limits unless the existing one has been proven guilty by the data! Make sure your baseline has not become some sort of moving average - it needs to stay fixed in time, even if that was three years ago.
Con - People can get over anxious to rebaseline. And, in the timeframe just following a trend, you don't have much data to make a new baseline from. This may also be a production line where you want to take corrective action to get the data back to the old baseline. Dr. Wheeler tends to default to don't change the baseline unless you have both a data shift, and you know why the data shift occurred.
Pro - "The job of management is prediction" (Deming). The baseline average and control limits provide prediction. If the current baseline has been proven no good, it is no longer a good prediction. We want to detect when the data stabilize out again, and set up new predictions.
In either case, SPC is a remarkable self-healing, self-correcting process. If you shift the baseline too early, on a false alarm, the new baseline will be proven guilty and you will be back on the old baseline.