I checked the Mobile Death Indexes for 1843 - 1875 at the library and could find no information on any Vaughn/Vaughan with any first names like Beckham or Hattie. I also checked the books on burials in Catholic Cemetery and Magnolia Cemetery with no results. Beckham is listed in the 1859 City Directory as a cotton factor - perhaps they moved on after 1860. You might want to check the port cities of New Orleans or Galveston TX.
Emma C Vaughan m William F Hunt, 12 Jan 1855 Mobile William F Hunt b 1834 MS d aft 1900 TX Emma C Vaughan b 1835 AL d aft 1900 TX
They are in the 1860 Mobile census with 2 children. In 1870 and 1880 they are in Lavaca Co TX censuses. In 1900 they are in Hays Co TX.
Their children appear to be: 1) Henry Beckham Hunt b 1857 AL (H B Hunt m M E Farrington, 8 June 1879, Lavaca Co TX.) They are in the 1880 Lavaca census as Becom & Elizabeth Hunt. 2) Maggie T Hunt b 1859 AL (possibly the Maggie F Hunt m A J Williams, 14 Dec 1879, Lavaca Co TX?) 3) William A Hunt b 1864 AL - does not appear in 1880 census, possibly died bet 1870 - 1880? 4) Albert D Hunt b 1865 TX 5) Martha Hunt b 1872 TX 6) Jessie Wilkins Hunt b 7 Mar 1876 Lavaca Co TX (Family Search has a record of her with her parents listed as William F Hunt & Emma Caroline Vaughan, although their DOBs are incorrect.) 7) Emma Hunt b 1879 TX