The blazar Mrk 501 still poses challenging questions for astrophysics, primarily because of the different orientations observed at pc- and kpc-scales. A much-debated issue is whether its core harbours a supermassive binary black hole (SMBBH). We remodel and reanalyse 83 data sets observed with the Very Long Baseline Array (2011-09-24─2023-07-24) at 43 GHz and compare them with earlier results for 15 and 8 GHz. In addition to the previously known parsec-scale jet in the core of Mrk 501, we detect a second nuclear jet (Jet 2), which appears to start off on the counterjet side and loop anticlockwise around the primary core. The image on 2022 June 24 indicates Jet 2 being lensed into a partial Einstein ring. The entire jet system (Jet 1 + Jet 2) at 43 GHz appears significantly displaced in a clockwise direction relative to the 15 GHz jet, and periodic transverse motion on a time-scale of about 7 yr is revealed. The total flux density light curve at 43 GHz also shows modulations with a similar period (7.4 yr). A shorter period of about 121 d is found in the 43 GHz flux density light curve of the core, albeit with less certainty. This shorter period is supported by the quasi-periodic appearance of Jet 2. We explore several physical scenarios and find that an SMBBH with a 7-yr orbital plane precessional period and a 121-d orbital period can reasonably explain our observational results. Assuming equal masses, we estimate the SMBBH to have its two components separated by 27─128 Schwarzschild radii and each having a mass of $10^{8}$─$10^{9}$ ${\rm M}_\odot$.