Sorry, there just isn't a good alternate route. There are only a few bridges over the river, and going over the mountains doesn't save time. There are only 2 roads into Chattanooga from the SW, I-24 and US-41. US-41 has 2 low RR underpasses, one is 13'1" the other is 10'4"
As for going around way south? You have to cross the river at Scottsborro AL. Scottsborro to Henager, Valley Head and Mentone. Down thru Summerville GA. US-27 south to GA-140 to I-75 north on I-75 to GA-136 at Resaca GA and east to US-411. Here you can go north to US-64 near Benton TN. Or keep going east to GA-515. Take 515 north up past Blue Ridge GA. Take GA-60 to Mineral Bluff and GA-60 spur to Ranger NC and US-64. Ranger is just west of Murphy NC
Edit: (West of Murphy NC)
