Personally I wouldn't go the focus stacking route for subjects as jittery as birds, especially small songbirds. It's not just their overall position but to successfully focus stack you'll want them to stay completely still including their heads throughout the whole focus stacking sequence which can be tough with the little guys.
Another way to look at this is to stop down for a bit more DoF and then if necessary slow down the shutter speed to keep your ISO where you want it. Yes, it's often not great to slow down shutter speed with small birds for the same reasons mentioned above, they tend to twitch a lot. But instead of say ten or twenty seconds of no motion on the bird's part you might only need 1/125" or 1/250" to account for stopping your lens down a stop or two while still keeping your ISO low which is much more likely than having a small bird stay perfectly still for many seconds so that each frame in the focus stack sequence aligns properly.
Also these days with the current crop of noise reduction tools, a bit more ISO usually isn't a show stopper unless you're really shooting at the edge of night.
Bottom line, focus stacks for live twitchy subjects is tough, focus stacking while shooting handheld is tough (remember the individual frames have to be well aligned before blending the stack) and most of the time you can compensate for shooting close and limited DoF by just stopping down a bit. If that doesn't drive ISO sky high then you're fine and if it does you may have to trade off some shutter speed but it still is generally a better choice than keeping your camera still and the hoping the subject stays still for many seconds. Relatively static wildlife subjects can be focus stacked but small birds may be quite a challenge.
