Just what is your destination? From Springfield to western Tennessee, I like to use US-60, to I-55 for destinations in eastern Arkansas or eastern Missouri, to US-51 for destinations in western Tennessee. US-60 is not limited access, but it is mostly four-lane divided. Shorter than dropping down to I-40, but not necessarily faster, still a lot faster than following US-412, which is even more direct but much slower traveling.
My route choices in that direction are all about avoiding the mess in West Memphis, but when really in a hurry, coming from Fort Smith (not Springfield) I will take I-40, do the extra miles, deal with the urban and suburban congestion, because the Interstate is faster, providing someone doesn't do something stupid to turn it into a parking lot.
From West Memphis, I've usually crossed into Tennessee on the I-40 bridge, as my destination would be Nashville or beyond. My time through, trying to connect to I-22, I used the I-55/US-61 bridge, found traffic at almost standstill on the Tennessee side, but if your destination is south side of Memphis, that is still the best way to go.
FWIW, I try to never get off the highway in West Memphis, the congestion on service roads is unbelievable. If I anticipate a fuel stop or rest stop along I-40, I will do that in Brinkley or Forrest City.
Unless the Memphis area stop is a necessity, from Springfield to Florida I would not go anywhere near Memphis. My Gulf Coast or Florida route involves taking I-40 to Little Rock, I-530 to Pine Bluff, US-65 to Lake Village, picking up US-82 to cross the Mississippi at Greenville. There are several routes off of 82 to the Gulf Coast, I like US-49 going from Indianola to Gulfport, but if actually headed toward Florida I would probably take US-98 from Hattiesburg to Mobile. Once again, these US numbered highways are mostly four-lane, mostly divided, but not limited access and you do have to slow down for some towns.