Through a two-stage transesterification/esterification and polycondensation method, three biobased poly(hexamethylene 2,5-furandicarboxylate-co-adipate) (PHFA) copolyesters with low contents of hexamethylene adipate (HA) units were synthesized in this research. The thermal properties, crystal structure, and tensile properties of PHFA copolyesters and poly(hexamethylene 2,5-furandicarboxylate) (PHF) homopolymer were extensively studied and compared with each other. All PHFA copolyesters still maintained good thermal stability. The glass transition temperatures, melting point temperatures, and equilibrium melting point temperatures of PHFA copolyesters gradually decreased with the increase of HA units. PHFA copolyesters displayed the same crystal structures as PHF. The elongation at break remarkably increased to 498.3 ± 12.0% for the PHFA copolyester containing 30 mol% of HA units (PHFA30) from 197.3 ± 12.2% for PHF; moreover, PHFA30 still showed a relatively high melting point temperature of 116.4 °C, similar to that of biodegradable poly(butylene succinate) (about 114 °C).