I am not a relative, just someone who lives in Shreveport, but my husband has a large collection of city directories, so I thought I would take a look. I came up with the following (I checked earlier directories, but 1909 was the earliest that had any possibilities for you):

1909 - Lejoy, M. (c), wks Spt Trac Co. (This would be the Shreveport Traction Company, which ran all the trolleys and streetcars)

1914 - Legie, Morris (c), lab, rms 219 Schumpert

1919 - Lejay, Morris (c), lab, rms 154 Western Ave

Lejay, Penny (c), laund, r 1919 Myrtle

1921 - Lejay, Penny (c), r. 1919 Myrtle

1922 - Lejay, Morris (c; Medora), lab, r. 1447 Hotchkiss

Lejay, Penny (c), dom, r. 1919 Myrtle

1923-24 - Lejay, Madora (c), r. 1816 Myrtle

1925-26 - No LeJays listed; however, a Penny Gaines is listed at 1919 Myrtle

1926-27 - LeJay, W.M., emp Va-Carolina Chem Co.

There are no LeJays listed in 1911 or 1913, and sadly I lack copies of 1910, 1912, 1915, 1916, 1917, 1918, or 1920.

There are several LeJays listed in the current Shreveport-Bossier telephone book.

Just a thought, but "Legier" is a not unusual Louisiana name and would be pronounced almost exactly the same way. I checked the directories for "Legier" and didn't come up with any luck, but it may help you fill in that large gap from the 1500s that you mentioned.

Hope this helps!
-SGB