Open-world knowledge graph completion (KGC) aims to infer novel facts by enriching existing graphs with external knowledge sources while maintaining semantic consistency under the open-world assumption (OWA). Generation-based KGC methods leverage the inherent strengths of large language models (LLMs) in language understanding and creative problem-solving, making them promising approaches. However, they face limitations: (1) The unreliable external knowledge from LLMs can lead to hallucinations and undermine KGC reliability. (2) The lack of an automated and rational evaluation strategy for new facts under OWA results in the exclusion of some new but correct entities. In the paper, we propose MusKGC, a novel multi-source knowledge enhancement framework based on an LLM for KGC under OWA. We induce relation templates with entity type constraints to link structured knowledge with natural language, improving the comprehension of the LLM. Next, we combine intrinsic KG facts with reliable external knowledge to guide the LLM in accurately generating missing entities with supporting evidence. Lastly, we introduce a new evaluation strategy for factuality and consistency to validate accurate inferences of new facts, including unknown entities. Extensive experiments show that our proposed model achieves SOTA performance across benchmarks, and our evaluation strategy effectively assesses new facts under OWA.