Where does it need more stablizing? In the deer itself or in the background? I would worry that additional background quilting will make your snowflakes not show anymore and that would be a shame. If in the deer, you can do short back and forth lines to simulate fur but then you run the risk of the deer being much more heavily quilted than the rest of the piece which presents a whole new set of issues.
Maybe you could try doing a few lines to represent the deer's musculature and a bit of shading in areas? I just did this in paint you can kind of see the few lines I drew in.
Edited to add, what you have shown so far is very pretty
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