In addition to teshuvah and tefillah, Klal Yisroel also increase tzedakah in the days of Aseres Yemei Teshuvah. It is discussed in אֵלִיָּה רַבָּה (סימן תקפא א) that the passage from Megillas Esther אִישׁ לְרֵעֵהוּ וּמַתָּנוֹת לָאֶבְיוֹנִים is roshei teivos אֱלוּל and the מִחִידָ"א says that צְדָקָה צָרִיך לַעֲשׂוֹת בְּיוֹתֵר בְּתִשְׁרֵי, וכמ"ש הָרַמְבַּ"ם דבעשי"ת מַרְבִּים בִּצְדָקָה during the עשי"ת one should increase acts of tzedakah.
Rabbeinu Efraim writes that the pasuk (ישעיהו א כז) וְשָׁבֶיהָ בִּצְדָקָה indicates that through giving tzedakah, one can merit teshuvah. One who gives tzedakah acts like a son of Hashem who is helping out his brother; forgiveness is an act that befits the relationship of a father to his children, as it says אָב שֶׁמָּחַל עַל כְּבוֹדוֹ כְּבוֹדוֹ מָחוּל when a father is ready to forgo the respect due to him by his children, they are indeed exempted. Rabbeinu Efraim finds a reference that this is a time to increase tzedakka in the pasuk (דברים טז כ) צֶדֶק צֶדֶק תִּרְדּף לְמַעַן תִּחְיֶה וְיָרַשִׁתָּ - Justice, justice shall you pursue, that you may live and possess the land Hashem, your God, is giving you. He explains that the word צֶדֶק can also mean צְדָקָה and the word יְרַשְׁתָּ is the same letters as תִּשְׁרֵי indicating that in Tishrei one should pursue acts of tzedakah and lovingkindness. Additionally, one cannot give tzedakah on four of the ten days of עשי"ת: Rosh Hashana, Shabbos and Yom Kippur. The six days remaining days in which we can give tzedakah are alluded to in the 1 of the word וִירַשִׁתָּ
Usually, the halacha is that one should not give more than a fifth of his earnings to tzedakah. The Tanya explains that this is in the case where one has not sinned, and is not looking for a way in which to repent. However, one who seeks repentance through giving tzedakah, is giving for his own sake. Just like a sick person would be willing to spend untold sums towards a cure, a person who is sick spiritually and is looking for a spiritual remedy may spend more than a fifth of his earnings on tzedakah.
The Tanya brings reference to this from the pasuk חַסְדֵי ה' כִּי לֹא תָמְנוּ the chessed of Hashem is without an end. When we are לֹא תָמְנוּ incomplete, we are not whole in our avodah, we must act like Hashem who does continuous and endless amounts of chessed with His children.