Soft Prompt Tuning (SPT) is a parameter-efficient method for adapting pre-trained language models (PLMs) to specific tasks by inserting learnable embeddings, or soft prompts, at the input layer of the PLM, without modifying its parameters. This paper investigates the potential of SPT for cross-lingual transfer. Unlike previous studies on SPT for cross-lingual transfer that often fine-tune both the soft prompt and the model parameters, we adhere to the original intent of SPT by keeping the model parameters frozen and only training the soft prompt. This does not only reduce the computational cost and storage overhead of full-model fine-tuning, but we also demonstrate that this very parameter efficiency intrinsic to SPT can enhance cross-lingual transfer performance to linguistically distant languages. Moreover, we explore how different factors related to the prompt, such as the length or its reparameterization, affect cross-lingual transfer performance.