We've previously discussed how Elul is a movement towards Tov. For a full explanation it is highly suggested that the reader see here.
We know that Moshe went up on Har Sinai on Rosh Chodesh Elul in order to receive the second Luchos, the second set of Tablet with the Ten Commandments. The Gemara in Brachos tells us that one of the defining characteristics of this second set was that, as opposed to the first this second set contained a reference to Tov - Goodness. It arises in the phrase "Lma'an Yitav Lach" 'So that it shall be good for you.'
In short, if the journey to receiving the "Tov" of the second Luchos begins on Rosh Chodesh Elul, then on some level the month of Elul as a whole is intrinsically attached to attaining Tov.
Perhaps we can add even more based on a teaching of the Arizal. We are taught that each one of the twelve months of the year parallel one of the twelve Shvatim, one of the twelve tribes of which the Jewish people are comprised.
The month of Elul goes along with the tribe of Gad. If we understand more about the arrival of Gad and its significance, then surely we will have a deeper understanding of the month of Elul.
When Gad is born the Passuk gives a very short statement about his arrival. "Bah Gad" 'Gad had come.' To explain this very vague line Rashi gives us two possible interpretations, each one with powerful links to the month of Elul.
The first explanation offered is that the term Bah Gad means "Bah Mazal Tov" 'Good fortune has arrived.' We with our previous understandings of this month the connection is beautiful. Elul is the month in which we attain Tov, and Elul, the month of Gad is an auspicious time in this regard, for we know that when Gad arrives - and his month - Bah Mazal Tov an air of Tov fills the world.
The second explanation given is that Gad was born Mahul, Gemalt, meaning he was born with his Bris Milah, he came out of the womb miraculously circumcised.
How is this connected to Elul? The four letters that comprise the word Elul are often broken apart as a part of several different acrostics, or as it is known in Hebrew, Rashei Teivos. In Parshas Eikev we are told, that Hashem will preform a circumcision of sorts on the hearts of the Jewish people. The Passuk says "U'Mol Hashem Elokecha Es Levavcha Ve'As Levav Zarecha." 'And God will circumcise the 'foreskin' (a reference to any impure blockage) of your hearts and the hearts of your descendants.'
Four words in this phrase, "Es Levavcha V'Es Levav" comprise the Roshei Teivos of the word Elul.
What emerges is that Elul is time when Hashem is more is excitedly waiting to help us remove the blockages, the Orla in whatever form that they take. Because if we start, if we simply desire, then Hashem will hopefully give us the Siyata Di'Shmaya that we need to overcome any obstacle and attach ourselves to Him.