Assuming that you meant Garmin 64st...

Looking at the specs online, that unit looks like it's a good high-tech handheld receiver with good circuitry and reception. So I don't think you're going to find anything more accurate in a handheld unit.

The fact that the GPS elevation data takes a while to settle down is troubling. Question: do you find that the elevation data settling down is correlated with its DOP readings? In other words, a better DOP means less movement of the elevation data?

Also does your GPS report its current VDOP number? The vertical precision indicator? Perhaps by digging into one of its configuration screens? If so, pay attention to that. If that number goes good and your altitude settles down, you probably have a "good enough" representation of altitude for your purposes.