The recently announced Super Wi-Fi Network proposal in United States is aiming to enable Internet access in a nation-wide area. As traditional cable-connected power supply system becomes impractical or costly for a wide range wireless network, new infrastructure deployment for Super Wi-Fi is required. The fast developing Energy Harvesting (EH) techniques receive global attentions for their potential of solving the above power supply problem. It is a critical issue, from the user's perspective, how to make efficient network selection and access strategies. Unlike traditional wireless networks, the battery charge state and tendency in EH based networks have to be taken into account when making network selection and access, which has not been well investigated. In this paper, we propose a practical and efficient framework for multiple base stations access strategy in an EH powered Super Wi-Fi network. We consider the access strategy from the user's perspective, who exploits downlink transmission opportunities from one base station. To formulate the problem, we used Partially Observable Markov Decision Process (POMDP) to model users' observations on the base stations' battery situation and decisions on the base station selection and access. Simulation results show that our methods are efficacious and significantly outperform the traditional widely used CSMA method.