There are many synonyms for the phrase "edge up," which typically means to slowly or gradually increase. Some alternatives include "rise up," "inch up," "creep up," "ascend," "climb," "escalate," "boost," "augment," "grow," and "progress." These synonyms can be used in a variety of contexts, from describing the growth of a company's sales to the slow increase in temperature during summertime. Additionally, each word carries slightly different connotations and shades of meaning, so choosing the right synonym can allow for more precise and evocative language.