I think so, and I try to prove it in this essay: “Was Joseph a Type of the Messiah? Tracing the Typological Identification between Joseph, David, and Jesus,” The Southern Baptist Journal of Theology 12.4 (2008), 52-77.
The gist of my article is this: From the reuse of key words and phrases (linguistic connections) and from parallels in significant event sequences (historical correspondence) we can see that the author(s) of the narratives concerning David in Samuel deliberately sought to point their readers to the narratives concerning Joseph in Genesis. Thus, the author(s) of Samuel saw Joseph as a type of David, and the two play similar roles in the outworking of salvation history. We find the same kinds of linguistic connections and parallels in event sequences between the narratives about Joseph and the narratives about Jesus, and Jesus fulfilled everything to which both David and Joseph pointed (escalation). Thus, Joseph was first a type of David, and then both Joseph and David were types of Jesus. In my judgment, this provides the necessary textual warrant to demonstrate both historical correspondence and escalation from Joseph through David to Jesus.
For the details, check out the essay: “Was Joseph a Type of the Messiah? Tracing the Typological Identification between Joseph, David, and Jesus.”
Here are my other attempts to exposit the typological interpretation practiced by the biblical authors in the Old and New Testaments:
“The Typology of David’s Rise to Power: Messianic Patterns in the Book of Samuel,” a Julius Brown Gay Lecture presented at The Southern Baptist Theological Seminary,” March 13, 2008.
“The Virgin Will Conceive: Typological Fulfillment in Matthew 1:18-23,” in Built upon the Rock: Studies in the Gospel of Matthew, ed. John Nolland and Dan Gurtner, 228-47. Grand Rapids: Eerdmans, 2008.