Quite a few commentators and thoughtful brethren correlate Paul's expression with the third cup of the traditional Passover meal. I don't, for a couple of reasons.

First, it is not entirely clear that Jesus ate a Passover meal in the traditional way at the traditional time with the disciples that night. Perhaps he did, but the gospel accounts are not crystal clear on this and it has been much debated for centuries, and is still debated.

Second, it is not certain just how Jesus or the Jews of Jesus' time celebrated Passover, and whether or not the traditions reported later were in vogue in Judea in Jesus' time. Sources for the traditions of first century (second temple, pre-Rabbinic) Judaism are meager. Did Jesus and his contemporaries have 3 or 4 named cups during the course of the Passover meal (if they ate the Passover that night)? We don't really know whether that custom existed at that time and place or not.

Third, the custom itself, where it does exist in Jewish tradition, is not a Biblical ordinance or example at all. There is no "cup of blessing" in the Old Testament, at Passover or any other time. The instructions for the celebration never included any instructions about wine or cups. The reference to the Passover in 1 Cor. 5:7 reflects on common Biblical knowledge, what the Bible actually said about it, and the same would not apply to the reference in 1 Cor. 10:16-17, nor does Paul make any direct connection to Passover in that passage. It is my understanding that there is also a Jewish custom (in post-temple Rabbinic Judaism) of having a "cup of blessing" at the close of the Sabbath meal, and at the conclusion of most holidays or special events. If Paul had any such traditional reference in mind, the conclusion of the weekly Sabbath would seem more appropriate in context, but again, that tradition is also not Biblical, and cannot be reliably documented to first century Judea or first century Judaism in the diaspora.

Either the Passover custom, or the more general Jewish custom relating to the "cup of blessing" may or may not have existed in Judea in Paul's time, and if it did it may or may not have been well known elsewhere. The uncertainty seems like quite a burden to place on Paul's comment. If he knew such a custom, what did it mean to him? Did he expect the readers in Corinth to know it? If he knew such a custom, and expected the Corinthians to know it too, what did he expect it to mean to them?

In terms of scriptural precedent, the closest Biblical analog that I know of for Paul's terminology is in Isaiah 65:8, which refers to God's future people as "new wine" in a preserved cluster "for it has a blessing in it."

Isa 65:8
8 Thus says the LORD:
"As the new wine is found in the cluster,
And one says, 'Do not destroy it,
For a blessing is in it,'
So will I do for My servants' sake,
That I may not destroy them all.
NKJV

• Vezot Haberakhah - THE CUP OF BLESSING - A rabbi's sermon on the general term