Did the northern nation of Israel or the southern nation of Judah date their reigns using years beginning in the fall, from the beginning of Tishri?

After the death of Solomon the nation of Israel divided into two: the northern kingdom ruled by Jeroboam, which was called Israel, and the southern kingdom ruled by Rehoboam, which was called Judah. The reign lengths and regnal data in I and II Kings seemed and seem entirely confused. No Bible student could arrive at a credible explanation of the data until the 20th century. The first one was Valerius Coucke: in the 1920s he came up with a solution to the problems of what appeared conflicting data in the reign lengths of the kings. Unfortunately, because he published in French, his work went largely unnoticed in the English speaking world. In the 1950s Dr Edwin Thiele published the fruits of his doctorate, "Mysterious Numbers of the Hebrew Kings". Later, Rodger Young slightly modified Edwin Thiele's results, in consequence of which the work of Valerius Coucke and ET/RY are in agreement, and both now place the Exodus from Egypt in 1446 BC. It is this consensus chronology which is used in Andrew Steinmann's "From Abraham to Paul - a Biblical Chronology". Both systems rely on interpretting some of the data so that reign-lengths are measured using years starting in Tishri, while other reign-lengths are measured from Nisan. My question is, what is the evidence that the recorders of the reigns in I and II Kings are sometimes using years beginning in Tishri? Does this evidence exist, or can the data be interpretted without needing to use years starting in Tishri?