The phrase "keeping close to" can be replaced with a variety of synonyms to express different shades of meaning. "Sticking with" or "clinging to" convey a sense of loyalty, commitment or adherence. "Adhering to" is more formal and implies compliance with rules or guidelines. "Following closely" suggests monitoring and observing, while "shadowing" implies close surveillance. "Maintaining proximity" or "staying within reach" suggest literal physical closeness, while "maintaining contact" or "keeping in touch" suggest ongoing communication. These synonyms can be used interchangeably, depending on the context of the sentence and the intended meaning.