“In New Orleans, you can live after a jam session if you didn’t have a good night,” Mario Abney says with a laugh when comparing the jazz scene in his previous home base of Chicago with the Crescent City’s. The trumpeter and composer, who moved to New Orleans in 2008 and just released his distinctive sophomore disc, The Abney Effect Vol. 1—Instant Grits, brings a bit of both of these major jazz cities to his sound.

“The musicians in Chicago are pressing for a certain type of express...