On a backpacking trip, calibrate the altimeter only at the trailhead and at places, such as a pass or a lake, that are marked on a topo map. In your comment you mention maybe calibrating the altimeter three times a day, and also by as little as 40 feet. This seems like overkill to me.
On a backpacking trip, we write down the altimeter reading when we stop for the day, and when we start for the day and maybe at the start and stop of a long break or flattish region if the weather looks like it is changing. If these notes are organized, it is easy to integrate mentally the changes caused by weather change rather than actual altitude change. (Yes, they are cumulative.) We calibrate the altimeter only when a significant change has accumulated -- 100 feet or so -- and when we are at a lake or pass or other benchmark.
We don't feel we need to know the altitude with great accuracy on a step by step basis. As for what change is significant for forecasting the weather purposes, it depends on the precision and accuracy of your altimeter. For a long time we had an analog altimeter which could be read to 10 feet precision, but which was not accurate to 10 feet. A rise or drop of 100 feet overnight was definitely significant, weatherwise. 50 feet, probably significant. (This is for the Sierra.)