The phrase "more deep-dyed" can be used to describe something that is deeply ingrained or strongly held. Some synonyms for this phrase include "firmly established," "entrenched," "ingrained," "set in stone," and "unshakeable." Other similar words include "inveterate," "indelible," and "deep-seated." These words suggest a strong and long-lasting commitment or belief. "More deep-dyed" can be used to describe a variety of situations, from deeply ingrained habits or beliefs to firmly established social structures or attitudes. Whatever the context, this phrase suggests a deep and lasting influence that is not easily changed or forgotten.