Texas has been described as part of the “South-west” since at least 1819.


(Oxford English Dictionary)
south-west, adv., n., and a.
U.S. The south-western states.
1835 J. H. OTEY in W. M. Green Mem. J. H. Otey (1885) 15 Connected with my journey to the South-West, was an ardent desire to forward..a projected plan of a Literary and Theological Seminary, to meet the wants of Episcopalians in Tennessee and Mississippi and Louisiana.

south-western, a. and n.
U.S. Of, pertaining to, or characteristic of the south-western states.
1806 New Eng. Palladium (Boston) 30 July 2/1 The President appoints the Legislative Councils in our South~western Territories.

10 September 1819, Daily National Intelligencer (Washington, DC), pg. 3:
Extract of a letter from an officer of the Republican army in the Province of Texas, to Col. R. Robinson, of this country.
“NACHITOCHES, July 12.
(...)
“The tide of emigration is bearing to the South-West like a torrent.”