I strongly believe that the wikipedia author has misquoted a german language reference. There are many places you can find reference to 25 (sometimes 27) altitude records beiong made by _other_ people, but the August 1932 flight to 53,000 ft seems to have been Piccard's last

>The following year, Piccard broke his record with an ascent to nearly 55,000 feet (16,764 m), and within a few years, **others** had risen to nearly 61,000 feet (18,593 m). Since then, manned balloons have risen to over 113,000 feet (34,442 m)  
<sub>Emphasis mine, source: https://www.encyclopedia.com/science/encyclopedias-almanacs-transcripts-and-maps/auguste-piccard-and-paul-kipfer-are-first-enter-stratosphere</sub>

A good breakdown of the 25 significant balloon altitude records can be found on [wiki](https://en.wikipedia.org/wiki/Flight_altitude_record) with the relevant section reproduced below. 

>- 1931-05-27: 15.781 km (51,770 ft) – Auguste Piccard and Paul Kipfer in a hydrogen balloon.
>- 1932: 16.201 km (53,150 ft) -Auguste Piccard and Max Cosyns in a hydrogen balloon.
>- 1933-09-30: 18.501 km (60,700 ft) USSR balloon USSR-1.
>- 1933-11-20: 18.592 km (61,000 ft) Lt. Comdr. Thomas G. W. Settle (USN) and Maj Chester L. Fordney (USMC) in Century of Progress balloon
>- 1934-01-30: 21.946 km (72,000 ft) USSR balloon Osoaviakhim-1. The three crew were killed when the balloon broke up during the descent.
>- 1935-11-10: 22.066 km (72,400 ft) Captain O. A. Anderson and Captain A. W. Stevens (U.S. Army Air Corps) ascended in the Explorer II gondola from the Stratobowl, near Rapid City, South Dakota, for a flight that lasted 8 hours 13 minutes and covered 362 kilometres (225 mi).
>- 1956-11-08: 23.165 km (76,000 ft) Malcolm D. Ross and M. L. Lewis (U.S. Navy) in Office of Naval Research Strato-Lab I, using a pressurized gondola and plastic balloon launching near Rapid City, South Dakota, and landing 282 km (175 mi) away near Kennedy, Nebraska.
>- 1957-06-02: 29.4997 km (96,784 ft) Captain Joseph W. Kittinger (U.S. Air Force) ascended in the Project Manhigh 1 gondola to a record-breaking altitude.
>- 1957-08-19: 31.212 km (102,400 ft) above sea level, Major David Simons (U.S. Air Force) ascended from the Portsmouth Mine near Crosby, Minnesota in the Manhigh 2 gondola for a 32-hour record-breaking flight. Simons landed at 5:32 p.m. on August 20 in northeastern South Dakota.
>- 1960-08-16: 31.333 km (102,800 ft) Testing a high-altitude parachute system, Joseph Kittinger of the U.S. Air Force parachuted from the Excelsior III balloon over New Mexico at 102,800 ft (31,300 m). He set world records for: high-altitude jump; freefall diving by falling 16 mi (26 km) before opening his parachute; and fastest speed achieved by a human without motorized assistance, 614 mph (988 km/h).
>- 1961-05-04: 34.668 km (113,740 ft); Commander Malcolm D. Ross and Lieutenant Commander Victor A. Prather, Jr., of the U.S. Navy ascended in the Strato-Lab V, in an unpressurized gondola. After descending, the gondola containing the two balloonists landed in the Gulf of Mexico. Prather slipped off the rescue helicopter's hook into the gulf and drowned.[a]


If Piccard would have broken his own record, I'm fairly certain it would have been documented in this list, but as you can see after the 1932 ascent the record was broken by others, with the US Navy breaking 23KM in 1956 - by which time Piccard had moved on to deep sea descent rather than stratosphere ascent

>In the mid-1930s, Piccard's interests shifted when he realized that a modification of his high-altitude balloon cockpit would allow descent into the deep ocean. 

Back to my original assertion, the final word will lie with whoever can check the reference from wikipedia:

> Dr. Erich Tilgenkamp - Reisen in ungewöhnliche Räume - Eine autorisierte Biographie - Verlag neues Leben Berlin 1956.