Here Linus is described as getting specific direction from the Apostle Peter on head covering. While only Paul writes about this matter in Scripture, Linus was a disciple of Peter, one of only three men who was specifically ordained as bishop by the Apostle. So he had the benefit of learning theology directly from him. So here we learn not only about Linus’ belief and practice, but that it was the apostle Peter’s too. This poses a problem for those that want to view Paul’s instructions in 1 Corinthians 11 as being only for that one specific church. If that’s so, why did the Apostle Peter instruct the bishops he trained and ordained to enforce head covering too? From this brief statement we learn a few other things regarding this practice. We see that head covering was not a suggestion, but a command. Linus is said to decree that a woman “must” veil. “Must” is the language of obligation. Second, we see that this is primarily a church issue. Linus’ decree was not concerning what a woman does at home, or in public, but how she comes “into the church.” This corresponds to Paul’s instruction that head covering is for when one is “praying and prophesying” (1 Cor 11:4-5) which shows that this symbol is specifically for how we are to worship.
This is from an Evangelical site, so the editors offer a disclaimer that they don't necessarily endorse all the claims of the article. Of course, it's difficult to see how one can use this information to uphold he apostolic tradition of headcovering, without also accepting the premise that St. Peter was the first bishop of Rome (pope) and that St. Linus was his successor.
I remember as an Evangelical using quotes of the Church Fathers to support specific positions, while ignoring everything else they said.