Jethro very plainly is advocating subsidiarity for Moses. I don't see any reference to distributism in that chapter though. Jethro is addressing organization/governance, not economics/ownership. (However, both terms are relatively new to me so I could be misunderstanding.)
There are several New Testament passages that assume subsidiarity without explicitly promoting it as Jethro does.
In Acts 6:1-6 the apostles realized that they were becoming overwhelmed with their responsibilities and directed the church to select the first deacons to take over some of the hands-on ministry responsibilities so that the apostles could devote themselves to prayer and to the ministry of the word (v4).
In Acts 13:1-3 the church at Antioch is functioning without direct oversight by the apostles in Jerusalem. Under the guidance of the Holy Spirit and without consulting the apostles, they send Barnabas and Saul on the first missionary journey. Later on, in Acts 15 there is a theological dispute that leads the church at Antioch to send Barnabas and Paul to consult directly with the apostles in Jerusalem for a decision.
One of the main purposes of the letter that Paul wrote to Titus was to instruct him in appointing elders/overseers in the churches that had already been established in Crete so that the churches could better function without the need for oversight from Paul or the apostles.