The Rambam writes in Hilchot Teshuva 3:4:
“Although Shofar blowing on Rosh Hashana is a divine decree, there is a hidden message of the Shofar. The message is for those who are spiritually asleep to awaken, carefully examine their behavior, perform Teshuva, and remember our Creator. Those who forget the truth in the course of daily routines and devote all of their time to temporal matters that have no lasting impact, should ponder their souls, improve their actions and thoughts. Everyone should abandon his evil actions and thoughts.”
The Rambam is writing about Rosh Hashana, but I don’t think he would object to us using the Shofar blowings of Elul as a wake up call.
