Despite recent advancements, the challenge of deep-dense tissue imaging with high resolution and contrast persists in breast cancer diagnosis; however, photoacoustic tomography (PAT) imaging addresses this issue by capturing both anatomical and functional information, including small masses as tiny as 3mm. Compressive sensing coined with PAT allows for a faster reconstruction, because it requires fewer sensors and samples. Although CS-PAT algorithms are efficient they rely on application specific parameter tuning and system matrix modelling. This paper introduces a time-efficient approach of algorithm unrolling, based on CS algorithms, to directly reconstruct high-resolution PAT images from sensor data, eliminating the need for explicit parameter assignment and system matrix modeling. The study proposes two unrolled deep learning networks based on split-Bregman total-variation and relaxed-basis-pursuit with ADMM implementation, followed by a denoising network for further enhancement. The model-free unrolled deep-learning approach successfully reconstructs high-resolution PAT images, even in the presence of noise, with low validation root mean square error. An enhancer network based on U-Net improves image quality to 0.91 and significantly reduces mean square error by 95%. Overall, the proposed algorithm unrolling method demonstrates promising potential for practical clinical applications, particularly in early disease detection, offering rapid image reconstruction without explicit system matrix modeling or parameter tuning. The inclusion of a U-Net denoising network enhances the approach's resilience and adaptability, suggesting possibilities for improved disease diagnosis and treatment outcomes, especially for early detection in dense tissues like breasts.