Unless you have any more specific information on your Grandfather's service it may be very difficult to trace the Battalion in which he actually served.
I believe over 20 Battalions of the HLI were raised during the First World War, each comprising between 600-1000 men. Of these Battalions, at least 10 fought during the first battles of the Somme offensive from July 1916 onwards. They are as follows:
a. 2nd Battalion, Highland Light Infantry, a
regular army formation which was brigaded under
the 5th Infantry Brigade, 2nd Division.
b. 157th (Highland Light Infantry) Brigade, 52nd
(Lowland) Division, a Territorial Force formation
comprising 1/5th (City of Glasgow) Bn, HLI, 1/6th
(City of Glasgow) Bn, HLI and 1/7th (Blythswood)
c. 10th and 11th (Service) Battalions, Highland
Light Infantry. These were known as Kitchener's
New Model Army Battalions and were raised from
August 1914 onwards. These units were
designated K1, indicating they were amongst the
first to form the 6 Divisions of the New Model
Army and were brigaded under 28th Infantry Brigade
in the 9th (Scottish) Division.
d. The 10th and 11th Battalions amalgamated to
form 10/11 Battalion HLI and was brigaded under a
K2 formation, the 46th Infantry Brigade, in the
15th (Scottish) Division along with the 12th
(Service) Battalion K2, Highland Light Infantry.
e. The 16th and 17th (2nd and 3rd Glasgow)
Battalions, Highland Light Infantry (K4)were
brigaded under 97th Brigade in the 37th Division.
f. Finally 1/9th Battalion, Highland Light
Infantry, a Territorial Force Unit, was initially
part of the regular 5th Infantry Brigade with
2HLI but was later re-allocated.
Sorry I can't be any more help