Lower, Mark A (1860) Patronymica Britannica: a dictionary of the family names of the United Kingdom. London: J.R. Smith. Public Domain.
We don't have any stats on how common this name is. This is probably because it's very rare in the UK.
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The Covert surname in historical dictionaries
Patronymica Britannica (1860)
COVERT. " Coverts," says Nelson, " are those woods which are thickets, and full of trees toucliing one another .... a covering or hiding-place for deer." Laws of Game. The great Surrey and Sussex family of Covert, whose contiguous manors are said to have extended from Southwark to the English Channel, traced their pedigree to temp. Henry II.