There are several synonyms for the term 'almost win' which can be used depending on the tone of the situation and context. The most commonly used ones include 'nearly win', 'close call', 'near miss' and 'fall short'. Each connotes a sense of coming oh so close to winning but ultimately falling short. Sometimes 'near victory' may also be used, but that is more focused on the achievement made rather than the non-achievement of winning. Regardless of which synonym is used, it's essential to use the correct one to convey the right message in the right context.