Massively multilingual language models such as multilingual BERT (mBERT) and XLM-R offer state-of-the-art cross-lingual transfer performance on a range of NLP tasks. However, due to their limited capacity and large differences in pretraining data, there is a profound performance gap between resource-rich and resource-poor target languages. The ultimate challenge is dealing with under-resourced languages not covered at all by the models, which are also written in scripts \textit{unseen} during pretraining. In this work, we propose a series of novel data-efficient methods that enable quick and effective adaptation of pretrained multilingual models to such low-resource languages and unseen scripts. Relying on matrix factorization, our proposed methods capitalize on the existing latent knowledge about multiple languages already available in the pretrained model's embedding matrix. Furthermore, we show that learning of the new dedicated embedding matrix in the target language can be improved by leveraging a small number of vocabulary items (i.e., the so-called \textit{lexically overlapping} tokens) shared between mBERT's and target language vocabulary. Our adaptation techniques offer substantial performance gains for languages with unseen scripts. We also demonstrate that they can also yield improvements for low-resource languages written in scripts covered by the pretrained model.