Instead of appealing to the obvious biblical antecedent of the potter analogy in Romans 9 (namely Isaiah 29:16 and 45:9) or the most logical intertestamental literature (namely Wisdom 15:7), many non-Calvinists will turn to Jeremiah 18.
Jeremiah 18:1-10
(1) The word which came to Jeremiah from the Lord, saying, (2) Arise, and go down to the potter's house, and there I will cause thee to hear my words. (3) Then I went down to the potter's house, and, behold, he wrought a work on the wheels. (4) And the vessel that he made of clay was marred in the hand of the potter: so he made it again another vessel, as seemed good to the potter to make it.
(5) Then the word of the Lord came to me, saying, (6) O house of Israel, cannot I do with you as this potter? saith the Lord. Behold, as the clay is in the potter's hand, so are ye in mine hand, O house of Israel. (7) At what instant I shall speak concerning a nation, and concerning a kingdom, to pluck up, and to pull down, and to destroy it; (8) If that nation, against whom I have pronounced, turn from their evil, I will repent of the evil that I thought to do unto them. (9) And at what instant I shall speak concerning a nation, and concerning a kingdom, to build and to plant it; (10) If it do evil in my sight, that it obey not my voice, then I will repent of the good, wherewith I said I would benefit them.
What non-Calvinists seem to overlook is the phrase "seemed good" in verse 4. The point in that passage is not "even if the clay pot turned out badly, the potter could still find something to use it for," rather it is that "the clay is malleable and God can make from the clay what it pleases God to make."
The translation "seemed good" is a valid translation but when we read it, we may overlook that this same phrase is used consistently to describe strong preferences:
Numbers 23:27
And Balak said unto Balaam, Come, I pray thee, I will bring thee unto another place; peradventure it will please God that thou mayest curse me them from thence.
Judges 14:3
Then his father and his mother said unto him, Is there never a woman among the daughters of thy brethren, or among all my people, that thou goest to take a wife of the uncircumcised Philistines? And Samson said unto his father, Get her for me; for she pleaseth me well.
Judges 14:7
And he went down, and talked with the woman; and she pleased Samson well.
1 Samuel 18:20
And Michal Saul's daughter loved David: and they told Saul, and the thing pleased him.
1 Samuel 18:26
And when his servants told David these words, it pleased David well to be the king's son in law: and the days were not expired.
2 Samuel 17:4
And the saying pleased Absalom well, and all the elders of Israel.
1 Kings 9:12
And Hiram came out from Tyre to see the cities which Solomon had given him; and they pleased him not.
1 Chronicles 13:4
And all the congregation said that they would do so: for the thing was right in the eyes of all the people.
2 Chronicles 30:4
And the thing pleased the king and all the congregation.
Jeremiah 18:4
And the vessel that he made of clay was marred in the hand of the potter: so he made it again another vessel, as seemed good to the potter to make it.
Jeremiah 27:5
I have made the earth, the man and the beast that are upon the ground, by my great power and by my outstretched arm, and have given it unto whom it seemed meet unto me.
The phrase combines yāšar (יָשַׁר) (straight or good) and ʿayin (עַיִן) (eye) (hence the translation at 1 Chronicles 13:4). While arguments from etymology should always come with an asterisk, the idea is that something pleasing is what your eyes settle or focus on. It's what you like to look at, as opposed to the things from which you turn away your face. We have the saying, "I can't take my eyes off of you," as an expression of desire, and it may convey some of the connotation of this particular expression. Another possible explanation for the expression could derive from the human love of symmetry and order, particularly to the eye. While some people revel in messiness, almost everyone has to concede that having things lined up properly is more pleasant to look at.
In short, the final stage of the pot per Jeremiah 18:4 is not up to the pot, nor is it a backup plan in case in the pot messes up the potter's first plan, rather it is what God in a very real sense wants to see.
While non-Calvinists will tend to emphasize the role assigned to nations (either to turn from evil or to turn to evil) and treat God as merely responsive to the nations, the outcome is nevertheless all in the hands of God. The point of the metaphor is not that clay gets to decide its own construction, but rather that God is free to do what God wants to do.
As a very minor side note, I found it interesting to see that the lexical form of the word for potter, yāṣar (יָצַר), differs from the lexical form of the word for straight/good, yāšar (יָשַׁר), primarily in that the former has a tzadie instead of a shin (yod-tsadeh-resh vs. yod-shin-resh). Thus, as reflected in the transliterations, the words sound a bit alike. I will leave it to others more versed in the pronunciation of Biblical Hebrew to say whether there is wordplay here, but I just found it interesting.