The Askaryan Radio Array (ARA) is a gigaton-size neutrino radio telescope situated near the geographic South Pole, which has been in operation since 2011. It is specifically designed to detect Askaryan emissions resulting from the interaction between ultra-high energy neutrinos ($>10$ PeV) and Antarctic ice. Each of the five ARA stations is equipped with 16 antenna clusters arranged in a cubic configuration, approximately ${\sim}200$~m deep in the ice. In this analysis, we utilize data from two ARA stations with a total livetime of ${\sim}10.5$ years, representing a two times increase in exposure time compared to the previous ARA result. To enhance the detection of neutrinos at the ARA, we introduce the template method known as a matched filter, which was inspired by analysis methods used by the Laser Interferometer Gravitational-Wave Observatory (LIGO) Collaboration. This method entails the direct comparison of data with simulated neutrino signal templates, facilitating the identification of low signal-to-noise ratio (SNR) neutrino signatures, ultimately resulting in an improved low energy threshold. Our study presents the diffuse neutrino search results derived from the analysis of data, with an estimated factor of 4 sensitivity improvements above 10 PeV. These improvements were achieved through a factor of 2.3 improvement in analysis efficiency due to the application of the matched filter technique and a factor of 1.7 increase in livetime.