Zero-shot translation is an open problem, aiming to translate between language pairs unseen during training in Multilingual Machine Translation (MMT). A common, albeit resource-consuming, solution is to mine as many translation directions as possible to add to the parallel corpus. In this paper, we show that the zero-shot capability of an English-centric model can be easily enhanced by fine-tuning with a very small amount of multi-parallel data. For example, on the EC30 dataset, we show that up to +21.7 ChrF non-English overall improvements (870 directions) can be achieved by using only 100 multi-parallel samples, meanwhile preserving capability in English-centric directions. We further study the size effect of fine-tuning data and its transfer capabilities. Surprisingly, our empirical analysis shows that comparable overall improvements can be achieved even through fine-tuning in a small, randomly sampled direction set (10\%). Also, the resulting non-English performance is quite close to the upper bound (complete translation). Due to its high efficiency and practicality, we encourage the community 1) to consider the use of the fine-tuning method as a strong baseline for zero-shot translation and 2) to construct more comprehensive and high-quality multi-parallel data to cover real-world demand.