This product was originally sold by John A. Perry and Milo S. Burr. Perry was only with Burr for a couple of years, but Burr continued to market the product. For more information, see John A. Perry.
The earliest advertisement for the Hungarian Balm I've found was in 1853 (view). According to Putnam, the product was also advertised in 1864. It was also advertised for sale by Frederick Baron in the 1856 Lowell Mass. Business Directory. Since no non-pontilled variants of the bottle are known, it is doubtful if it was for sale much later than that. Another possibility is that they may have changed to a label-only bottle. After looking ad ads, there must have been 4 bottle sizes.
Rectangular, 5 3/4", graduated top, front & side panels, O.P., aqua, side: "PERRY'S" front: "HUNGARIAN - BALM" side: "FOR THE HAIR"