Provisioning reliable network services for mobile users in a mobile edge computing environment is the top priority for most network service providers, as unreliable or severely failed services will result in tremendous loss on their revenues and consumers. In this paper, we study a novel service reliability augmentation problem in a Mobile Edge-Cloud (MEC) network, where mobile users request various network services through issuing requests with service function chain (SFC) requirements and reliability expectations, and an admitted request may not meet its reliability expectation initially. To enhance its service reliability to reach its expectation, it is a common practice to make use of redundant backups, that is to place redundant VNF instances of each Virtual Network Function (VNF) in its SFC in case its primary VNF instance fails. In this paper, we aim to augment the reliability of each admitted request as much as possible with the ultimate objective to reach its reliability expectation, subject to computing capacity on each cloudlet in the network. To this end, we first formulate a novel service reliability augmentation problem. We then deal with the problem for the admitted request under the assumption that all the secondary VNF instances of each primary VNF instance in its SFC must be placed into the cloudlets no more than l hops from the cloudlet of the primary VNF instance, where 1 ≤ l ≤ n - 1 and n is the number of cloudlets in the network, for which we propose an integer linear program (ILP) solution, and develop a randomized algorithm with a provable approximation ratio while a moderate resource constraint violation. We also devise an efficient heuristic algorithm for the problem without any resource constraint violation. We finally evaluate the performance of the proposed algorithms through experimental simulations. Experimental results demonstrate that the proposed algorithms are promising, and their empirical results are superior to their analytical counterparts.