Question: Does Hebrews 13:20 refer to the New Covenant, and if so wouldn’t that provide evidence that the church is currently under the New Covenant today?

Answer: Not necessarily, but possibly (and probably) and…no.

Hebrews 13:20 does not directly refer to the New Covenant (NC). The descriptor is that the covenant is eternal (αἰωνίου) and not new (καινήν). Because of that, the burden of providing exegetical evidence would be on the one who would assert that the reference is to the NC (I think a strong theological case could be made, but not necessarily an exegetical one). Still, the passage can most certainly be a reference to the NC, but it is not what I would call a direct reference, simply because a different qualifier is used, and the resulting possibility (though not probability) that something else may be in view..

Nonetheless, it is certainly possible – and maybe even probable –  that Hebrews 13:20 does refer to the NC. Regardless of what covenant is intended, the passage indicates that He is the Shepherd of the sheep through the blood, not the covenant (blood is dative [or instrumental], covenant and eternal are genitive). In the context of the Lord’s Supper, the believer’s activity is to remember Him, and to proclaim His death, not the covenant  (1 Cor 11:25-26). In Hebrews 13:20, the blood  – not the covenant itself – has accomplished something significant. The covenant is the promised occasion for the shedding of the blood, and is obviously not unimportant (in fact it is absolutely vital in God’s revealed plan), but the covenant is a separate agreement that accomplishes a specific stated purpose outlined with detail in Jeremiah 31. The author of Hebrews utilizes the NC as one of several great evidences of Christ’s superiority and therefore His worthiness to be trusted; nowhere does the author of Hebrews place the church under the covenant itself. (Nor does any other NT writer, in my estimation).