Oh, you're talking about alleles, not linkage.
The locus is the place on the chromosome where the gene is.
Alleles are the different forms of the gene.
At the Dominant White locus, there are at least 3 alleles (Dominant White, dun, wild-type). The e-locus has at least 5 alleles (Extended Black, Birchen, Wheaten, Partridge, wild-type)). The Barring locus is already known to have several different alleles, that cause various amounts of white.
But no chicken can have more than 2 alleles at a given locus: one from the father, and one from the mother. A hen cannot have more than 1 allele for any sex-linked gene, because she only has one Z chromosome.
So for your hen, she would not inherit anything on the Z chromosome from her mother. Her father was barred, so one of his Z chromosomes had barring. She is not barred, so her Z chromosome does not have barring. That tells us what her father's second Z chromosome has: not-barring (or maybe some weird form of barring.)
Does this hen have any sisters from the same father? If there are any normal not-barred ones, then the father has a normal not-barred allele on his Z chromosome, and that's what this hen inherited too.