My family is in possession of all tellegrams related to his injury, no exact date of death can be established but the last information ever found about him is as follows. (pardon the copy paste formatting)

I do know from correspondence we have that Byrons family cared for Milton until death, and the scarcity of information about him is because most are looking for information on Milton, when he was most commonly known as Bud even in legal texts. The second reason was that Susan and Tilford had a run of illness that killed almost the entire family, the rest being orphaned out.

Newcastle News Journal
Jan 16, 1903
Milton Lamb, a sheepherder in the employ of W. H. Minter
at Upton, had both his feet frozen last Saturday. He was brought to
this city for medical aid and is under the care of the doctors at the
Newcastle house.
Newcastle News Journal
Jan 30, 1903 Friday
Feet Amputated.
Wednesday evening Dr. Horton amputated both feet of Bud
Lamb, the unfortunate sheepherder who had them severely frozen some
three weeks ago. The operation was a successful one and the patient is
getting along nicely at this writing. The limbs were amputated above
the shoe tops.
The Laclede Blade
Saturday Feb 7, 1903
Front Page, Col 6 part way down
Loses Both Feet
Byron Lamme has received word that his brother, Milton,
generally known here as "Bud", had the misfortune to get both his feet
frozen on January 10 and in an effort to save his life both members
were amputated just below the knees last week. He is in Wyoming. Mr.
Lamme expects to leave for Wyoming in a few days and bring his brother
home as soon as he is able to come.
News Letter Journal - Newcastel, Wyoming
March 13, 1903
Milton Lamb the unfortunate sheepherder who had both feet frozen
and amputated, was sent to LaClede, MO. Wednesday evening, where he
will visit with a brother. "Bud" went away feeling "mighty fine"
notwithstanding the hardships and trying times he has had within the
past two months.
The Laclede Blade
Saturday March 21, 1903
Front Page, Center Col bottom
Milton Lamme, who had the misfortune to lose his feet from
freezing, arrived here from Wyoming last week and is now being cared
for at the home of his brother, Byron, north of town. The wounds are
said to be healing nicely and he is otherwise enjoying good health.