There is no single right answer for modernizing Assembly / HLASM. Here is what the realistic paths actually look like, and what matters most when choosing between them.
Assembly is the one language on this site where a full rewrite is genuinely rare. Code written in HLASM usually exists because it needs direct control over registers, storage, or performance that a higher-level language cannot provide, which means the realistic modernization path is almost always leaving it in place and building modern interfaces around it, not replacing it. Budget for skilled review time instead: the risk is not the language, it's the small number of people left who can read it.
Rewrite or convert to a new platform
Vendors taking this approach for Assembly / HLASM: Astadia, TSRI, Modern Systems (OneAdvanced), MicroAPL, Semantic Designs.
Keep it running, don't change it yet
Vendors taking this approach for Assembly / HLASM: IBM.