$ε$-Minimum Storage Regenerating ($ε$-MSR) codes form a special class of Maximum Distance Separable (MDS) codes, providing mechanisms for exact regeneration of a single code block in their codewords by downloading slighly sub-optimal amount of information from the remaining code blocks. The key advantage of these codes is a significantly lower sub-packetization that grows only logarithmically with the length of the code, while providing optimality in storage and error-correcting capacity. However, from an implementation point of view, these codes require each remaining code block to be available for the repair of any single code block. In this paper, we address this issue by constructing $ε$-MSR codes that can repair a failed code block by contacting a fewer number of available code blocks. When a code block fails, our repair procedure needs to contact a few compulsory code blocks and is free to choose any subset of available code blocks for the remaining choices. Further, our construction requiresa field size linear in code length and ensures load balancing among the contacted code blocks in terms of information downloaded from them for a single repair.