The tachi was still in use as a cavalry sword. For example, in this picture of the Siege of Osaka Castle, 1615:
the cavalry wear tachi, while most infantry wear a katana (some armoured infantry wear tachi, too, like the musketeer near the gate at the bottom left, and the archer at the bottom extreme right).
As late as the Satsuma Rebellion, art still shows traditionally-equipped armoured cavalry wearing tachi (and modernised cavalry with Western sabres, modernised infantry with katana worn in baldrics, unarmoured samurai women fighting on horseback with katana - lots of variety).