National Competence Center (NCC) Türkiye has created the initial structure comprising national High-Performance Computing (HPC) centers, academia, and partnerships to increase Small and Medium-Sized Enterprise (SME) HPC capabilities toward the developing European HPC (EuroHPC) ecosystem. The NCC has started 29 collaborations with SMEs and produced success stories in the manufacturing, transportation, social media, and engineering sectors. NCC Türkiye has planned to upscale the HPC maturity of SMEs for emerging European supercomputers. Our experiences have shown that SMEs need a tool to comprehensively assess their current situation regarding HPC capabilities and assist them in identifying and prioritizing reasonable improvement opportunities. Hence, we have developed the SME HPC Maturity Model (SME-HPC-MM) based on our SME collaborations, EuroCC definitions, and Information Technology (IT) process assessment standards. The maturity model has a structure that supports SMEs' participation in the EuroHPC ecosystem. Furthermore, the model can improve the NCC-SME collaborations by guiding process improvements in SMEs for developing HPC capabilities.