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We construct an explicit linear strand for the initial ideal of any determinantal facet ideal (DFI) with respect to any diagonal term order. We show that if the clique complex of Δ has no 1-nonfaces larger than a certain cardinality, then the Betti numbers of the linear strand of and its initial ideal coincide. This confirms a conjecture of Ene–Herzog–Hibi for closed graphs with at most 2 maximal cliques. Additionally, we show that the linear strand of the initial ideal of any DFI is supported on an induced subcomplex of the complex of boxes introduced by Nagel–Reiner.