Do you know of any?
I searched for a first-hand account of Coronado’s and came up empty as well.
As far as the DeSoto Chronicles - it’s a compilation of several authors who wrote of the expedition after the fact.
These books are recent translations of the four major accounts of the De Soto expedition - the Gentleman of Elvas, Luys Hernandez de Biedma, Rodrigo Ranjel, and Garcilaso de la Vega, based on the account of the soldier Gonzalo Silvestre.
I dont believe it would truly qualify as a ‘first-hand account’ except there are short portions included from those who participated.
The 2 volume set I acquired from Amazon -
https://www.amazon.com/Soto-Chronicles-Expedition-Hernando-1539-1543/dp/0817308245