For another reconstruction of the historical Saul from the literary Saul, see Diana V. Edelman, “Saul ben Kish in History and Tradition,” in The Origins of the Ancient Israelite States, ed. Volkmar Fritz and Davies, JSOT Sup 228 (Sheffield: JSOT, 1996), pp. 142–159.
Whether Saul gained the throne immediately after the battle of Jabesh-Gilead or later is another matter. Gilgal is quite distant from Jabesh-Gilead, and it would have been a long and grueling march. A closer shrine at which to perform the inauguration, such as Shechem, could have easily been found. I believe it is more likely that Saul’s army made him king at Gilgal after he led them to victory against the Philistines.
If that is true and Saul was made king by the grateful Benjaminites and Ephraimites shortly after the battle with the Philistines at Michmash, then the location of his inauguration at Gilgal makes perfect geographical sense. Gilgal is on the border between the two tribal areas and would have been convenient to both. More importantly, holding the inauguration at a location between the two tribes would have signified the binding of the two as subjects of Saul.