In his book on Gen 14:18-20 and its echoes throughout the Torah, Joshua G. Matthews forwards the thesis that:
[T]here is a textually recognizable and demonstrably distinct priestly succession—an order of Melchizedek—intended in the composition of the Pentateuch and continuing throughout the OT canon (Tanak). (Joshua G. Matthews, Melchizedek’s Alternative Priestly Order: A Compositional Analysis of Genesis 14:18-20 and Its Echoes Throughout the Tanak [Bulletin for Biblical Research Supplement 8; Winona Lake, Ind.: Eisenbrauns, 2013], 2-3)
And that:
The first matter to consider is the portrayal of Aaron in the Pentateuch. I am suggesting that Melchizedek initiates a priestly succession, or order, meant to be seen as an alternative priesthood to that of Aaron and his successors. (Ibid., 80)
On Exo 18 and the depictions of Aaron and Jethro and how they relate to his thesis, Matthews notes:
The result is an apparent contrast between Jethro and Aaron. These contrastive features may be presented in three groups.
The first group. After Moses and Jethro greet each other, Moses tells Jethro of all Yahweh had done to Pharaoh for Israel, the hardships they had encountered along the way, and how Yahweh had rescued them (וַיַּצִּלֵ֖ם, vv. 7-8). Jethro responds by rejoicing about all the good that Yahweh has done for them about how he rescued them from the hand of Egypt (הִצִּיל֖וֹ מִיַּ֥ד מִצְרָֽיִם, v 9). This joyous declaration leads to Jethro’s proclamation of blessing to Yahweh in which he repeats multiple times the language of rescue from the hand of Egypt and Pharaoh: “Blessed be Yahweh, who rescued you from the hand of Egypt and from the hand of Pharaoh, who rescued the people from under the hand of Egypt” (הִצִּ֥יל אֶתְכֶ֛ם מִיַּ֥ד מִצְרַ֖יִם וּמִיַּ֣ד פַּרְעֹ֑ה אֲשֶׁ֤ר הִצִּיל֙ אֶת־הָעָ֔ם מִתַּ֖חַת יַד־מִצְרָֽיִם, v.10). This repetitious mention of rescue (נצל) recalls Yahweh’s promise in 3:8. There, when he first called Moses, Yahweh had said that he had heard the cries of the Israelites and had come down to rescue them from the hand of the Egyptians (לְהַצִּיל֣וֹ׀ מִיַּ֣ד מִצְרַ֗יִם). Perhaps also in mind is Moses’ faithless and emphatic protest that Yahweh indeed did not rescue (וְהַצֵּ֥ל לֹא־הִצַּ֖לְתָּ) the people (5:23). Whereas Moses earlier had disparaged Yahweh for failing to rescue Israel when and in the way that he wanted, and whereas nowhere yet have we read of Moses, Aaron, or any Israelite expressing thanksgiving to Yahweh for rescuing them from Egypt, here we see a non-Israelite priest rejoicing and blessing Yahweh for his miraculous rescuing acts.
The second group of features contrasting Jethro and Aaron: Jethro is presented as a more pious priestly figure than Aaron because of Jethro’s knowledge of Yahweh. In Exod 18:11, just after blessing Yahweh for rescuing the people from Egypt, Jethro says, “Now I know (יָדַ֔עְתִּי) that Yahweh is greater than all the gods.” It does not seem to be a coincidence that this statement reflects language repeated at pivotal moments earlier in the exodus narrative. Multiple times Yahweh had said that his signs would help either the Israelites (6:7; 16:6, 12) or Pharaoh and the Egyptians (5:2; 7:5, 17; 8:6, 18; 9:14, 29; 11:7; 14:3, 18) to know him as Yahweh. Eugene Carpenter, in his essay on the structural function of chapter 18 within Exodus, points out the significance of Jethro’s response to Yahweh’s acts as it relates to the theme of knowing in the earlier chapters of Exodus. In particular, Carpenter regards Pharaoh’s response to Moses in 5:2 as programmatic of this motif. There Pharaoh asks, “Who is Yahweh?” and says, “I do not know Yahweh!” when Jethro declares, “Now I know,” this statement is intentionally presented in stark relief to Pharaoh’s earlier “I do not know” (Eugene E. Carpenter, “Exodus 18: Its Structure, Style, Motifs and Functions in the Book of Exodus,” in A Biblical Itinerary: In Search of Method, Form and Content—Essays in Honor of George W. Coats [ed. Eugene E. Carpenter; Sheffield: Sheffield Academic Press, 1997], 103). Moreover, whereas the narrative never has either Moses, Aaron, or the Israelites “knowing Yahweh” as a result of the miraculous deliverance from Egypt, here Jethro clearly does know.
The third group of features contrasting Jethro and Aaron: Jethro offers a burnt offering and sacrifices, eliciting thoughts of priestly functions. In this way, Jethro fulfills what Yahweh had instructed Moses to tell Pharaoh: that they were going to sacrifice to Yahweh in the wilderness (Exod 5:3). In addition to Yahweh’s initial instructions to Moses, these sacrifices were mentioned so often that, by this point in the narrative, we as readers wonder when they will come to fruition (cf. 5:8, 17; 8:4, 21-25; 10:25; 12:27; 13:15). Now Jethro, the Midianite priest, carries them out (see Carpenter, “Exodus 18,” 101-2). These sacrifices and burnt offering precede the revelatory institution of the cult at Sinai, and it is significant that they are carried out by a priest other than Aaron—he is even a non-Israelite. Though Jethro and Aaron are not directly compared, Jethro is portrayed as fulfilling this fundamental priestly function prior to Aaron’s official establishment as priest.
Again, these nuanced features do not directly contrast Jethro and Aaron as individuals. Nevertheless, at the very least, we do see in the details of Jethro’s interactions with Moses, Yahweh, and the Israelites at the foot of Sinai the portrait of a non-Israelite priest who embodies faithful and laudable priestly activity. Moreover, given Aaron’s negative role earlier in Exodus, the direct verbal association between Jethro and Aaron in their greetings of Moses, and that of Aaron and his priesthood move into the narrative spotlight in the expansive central portion of the Pentateuch that begins just one chapter later, Jethro does appear to be presented in Exodus 18 in compositional juxtaposition over against Aaron. This, I am suggesting, contributes to the notion of an alternative priesthood to Aaron’s. (Ibid., 99-101, emphasis added)
Such comments should pique the interest of Latter-day Saints, showing that there was a genuine God-ordained priesthood, after the likeness/order of Melchizedek, operative during the time of Moses and others. Indeed, as I note in my book on the priesthood, Jethro, who is referred to as a כֹהֵ֖ן held the Melchizedek Priesthood and such also supports the revelations of Joseph Smith which state that Jethro was a (true) priesthood holder who ordained Moses to such (D&C 84:6-7). (see After the Order of the Son of God: The Biblical and Historical Evidence for Latter-day Saint Theology of the Priesthood [2018]).