Sorry for the delay,
`That is strange. Usually if there is a scaling issue it has to due with the bind-pose (the resting pose of the skinned mesh). But this sounds like there might be some else going on. Can you inspect the rig a see if the transforms look normal I suspect there could be a root bone scaled to say 100 rather than 1 which could look normal if the child bones where at a 0.01 but might cause some issues. If that is the case you can unparent the child bones a set the scale of the root to 1 and reparent the child bones.
If the transforms are not the issue you can try the Edit/Skin/Bindpose Command and if that does not work can you send me a unity package to debug the issue.