That the Bible affirms that more than Jesus and Melchizedek (who is not an Old Testament “Christophany”) hold the Melchizedek Priesthood can be shown by the fact that the Davidic Kings were the addressee of the following text:
The Lord has sworn and will not change his mind, "You are a priest forever according to the order of Melchizedek." (Psa 110:4 NRSV)
This would explain why David and Solomon, who were not Levites, would engage in priestly sacrifices and other actions (e.g.,2 Sam 6:12-14 [cf. Exo 28:6]; 2 Chron 6:13; 1 Kgs 8). Furthermore, the term כֹּהֵן (priest [KJV: rulers]) is used of the sons of David in 2 Sam 8:18. Here is the NRSV translation of this verse:
Benaiah son of Jehoiada was over the Cherethites and the Pelethites; and David's sons were priests (כֹּהֲנִ֥ים).
All this adds support to the Latter-day Saint contention that other people than Jesus and Melchizedek held the Melchizedek Priesthood (cf. D&C 107).
To understand the desperation many critics engage in to avoid the obvious claim that non-Levites served as priests in the Old Testament can be seen in this thread on the Mormon Dialogue forum. With respect to 1 Sam 2:18 ("But Samuel ministered before the Lord, being a child, girded with a linen ephod"), one Evangelical wrote the following (lame) argument:
1 Samuel 2:18 does not say that God called Samuel to serve as a priest. The passage says that Samuel worked as a servant to a priest, not that he was a priest himself (see v. 11). It was natural enough for assistants of the priests to wear a linen ephod, but this didn't make those assistants priests themselves.
Responding to this, Bill Hamblin wrote the following:
How about the fact that Moses and Samuel are equated in Jer. 15:1.
Moses and Aaron were among his priests,
Samuel was among those who called upon his name. (Ps 99:6)
Here, in Hebrew poetic parallelism, Samuel is a priest like Moses.
What about Samuel offering burnt offerings (1 Sam. 7:9)?
So:
1- he wears priestly robes (1 Sam 2:18)
2- he "serves" in the temple/tabernacle (1 Sam 2:18), a technical term for temple liturgy (HALOT 1661-2)
3- he offers sacrifice (1 Sam 7:9)--supposedly a prerogative of priests.
4- he, like Moses is "stands before the Lord" (Ps 99:6 and Jer. 15:1)
The obvious conclusion is that Samuel was a priest, though not a Levite. Alas, since it doesn't match your Evangelical theology, you reject the obvious meaning of the text.
One can read the further exchange on this point on the thread, but it does show that Evangelical Protestants only play lip-service to follow the plain meaning of the biblical texts.