Why at this juncture (posuk 15, perek 13) did Moshe call Hoshea, Yehoshua when Hashem had previously told Moshe (Parshas Beshalach posuk 14, perek 17) to remind YEHOSHUA of the continual fight against Amalek and the Torah calls Yehoshua by that name on many occassions prior to Parshas Shelach Lechah? 

The Rosh says that Moshe "sensed" that Yehoshua would "inherit" the reward that should have been given to the other 10 spies. Thus the letter YUD (the numerical value of 10) was added to Hoshea's name. So the reason why he was called Yehoshua in now explained.

The Gra contends that Moshe DID NOT change Yehoshua's name but Moshe prayed that Hashem (Yud and Heh) should save him - Hoshea from the machinations of of the spies. The Gra says that is why the Torah refers to Hoshea by that name in Parshas Ha'azeenu posuk 44 Perek 32 : he had 2 names!!