John Adams had been readmitted to hospital with a recurring infection and inflammation of the wound as indicated in his letter of 14th June.
Records from are available from No 3 Casualty Clearing Station, then at Puchevillers behind the lines at Hamel. See https://www.forces-war-records.co.uk/records/7869695/corporal-j-adams-royal-irish-rifles/
These records indicate the following
Index number of admission: T1051
Ailment: S[ick]. Inflammation of connective tissue, hand
Date of Admission for Original Ailment: 17/06/1916
Date Transferred to Sick Convoy: 18/06/1916
Number/Designation of Ward: A1
Notes written in the Observations Column: 110th Field Ambulance. To No.18 Ambulance Train.
110th Field Ambulance was part of 36th Ulster Division, and in June 1916 was located at Clairfaye Farm between Varennes and Léalvillers.
For more on No 18 Ambulance train see http://www.warwickshirerailways.com/misc/misc_equip246.htm
Archive Reference: MH106/311 can be found at The National Archives in Kew, and contains First World War Representative Medical Records of Servicemen from No. 3 Casualty Clearing Station