Well, in Exodus 34:14 Israel is already being warned not to "whore after their gods." By describing idolatry with the language of prostitution and sexual immorality, Moses is talking about the covenant between Yahweh and Israel as though it is a marriage. So this way of thinking about God's relationship with his people is well established prior to the time of Solomon, and it continues after Solomon, not least with Hosea, where when Hosea marries Gomer, Hosea plays the part of Yahweh, Gomer the part of Israel. 

So I think we can be confident that biblical authors prior to and after Solomon were thinking about a spiritual meaning of marriage, recognizing an analogy between human marriage and God's covenant with Israel. Is there more specific evidence? 

Psalms 45:1 is perhaps the closest analogy to the Song of Songs in the Old Testament, being a wedding song for Israel's king. The Psalm begins with a celebration of the king in Psalm 45:1-9, then concludes with an address to the princess marrying the king in 45:10-17. As the psalmist extols the greatness of the king, he says in Psalm 45:6, "Your throne, O God, is forever and ever," showing—at least—a very tight connection between Yahweh and the king who represents him. With a statement like this in Psalm 45, and with other texts in the OT communicating a very close connection between Israel's God and Israel's king (e.g., Numbers 23:21; Isaiah 9:6; Jeremiah 23:6; Hosea 3:5; Micah 2:13; Micah 5:4; Zechariah 12:8; Zechariah 13:7) it would seem natural—not forced or fanciful—to see an analogy between the King and his Bride and Yahweh and Israel in the Song of Songs. 

What about evidence that later OT authors might have read the Song this way? Is there anything that points in that direction? The King in the Song is regularly called the Bride's "beloved." This particular Hebrew word means different things in different contexts. In some contexts it means "uncle." It is not often used outside the Song of Songs the way Solomon uses it in the Song. In fact, the only place outside the Song of Songs where the word is used with the same meaning it has in the Song is Isaiah 5:1, where Isaiah writes, "Let me sing for my beloved my love song concerning his vineyard." What follows in Isaiah 5 makes it clear that Isaiah is referring to Yahweh as his beloved. Given the fact that the only Scripture in which the word is used this way prior to Isaiah is the Song of Songs, it would seem at least possible that Isaiah's thinking about the Lord has been influenced by the Song, with the result that Isaiah refers to the Lord the way the King is referenced in the Song. A related form, though not exactly the same Hebrew word, is used in a similar way, with reference to the relationship between Yahweh and Israel, in Jeremiah 11:15, Ezekiel 16:8, Psalms 60:5/Psalms 108:6, and Psalms 127:2