As the Internet becomes a critical infrastructure component of our global information-based society, any interruption to its availability can have significant economical and societal impacts. Although many researches tried to improve the resilience through the BGP policy-compliant paths, it has been demonstrated that the Internet is still highly vulnerable when major failures happen. In this paper, we aim to overcome the inherent constraint of the existing BGP-compliant recovery schemes and propose to seek additional potential routing diversity by relaxing BGP peering links and through Internet eXchange Points (IXPs). The focus of this paper is to evaluate the potentiality of these two schemes, rather than on their implementations. By collecting most complete AS link map up-to-date with 31K nodes and 142K links, we demonstrate that the proposed potential routing diversity can recover 40% to 80% of the disconnected paths on average beyond BGP-compliant paths. This work suggests a promising venue to address the Internet failures.