Avery won her first senior national title in 2014, winning over 1500 m at the English Athletics Championships. She also paced sixth in the 5000 m at the British Athletics Championships, having improved her best to 15:35.12 minutes earlier that season. She began to progress in the under-23 category the following year, starting with a win in the 10,000 metres at the Penn Relays. She won the 5000 m at the under-23 trials before going on to take the bronze medal in that event at the 2013 European Athletics U23 Championships, which was later upgraded to silver. She enrolled at Iona College in the United States towards the end of the year and began competing for their Iona Gaels collegiate team. Her first collegiate honours quickly followed at the 2013 NCAA Division I Cross Country Championships, where she paced third. A month later she shared in the under-23 team title with Great Britain at the 2013 European Cross Country Championships, having finished fourth behind team mate Charlotte Purdue.