For faster than Mach 3.5, the answer is unfortunately no until Virgin Galactic or one of their would-be competitors makes it happen. Suborbital flights like those planned by Virgin Galactic seem most likely to be the first thing to top that number on a commercially-available, recurring basis.
The fastest non-rocket-powered manned vehicle so far was the SR-71 Blackbird. Officially, it got up to around Mach 3.3, though SR-71 pilot Brian Shul (same guy who famously tells the LA speed check story) claimed to have briefly exceeded Mach 3.5 while evading a missile over Libya in his book. Needless to say, rides in SR-71s were never commercially available and the vehicle is retired completely nowadays. If you were going to exceed Mach 3.5 in something other than a rocket, though, the SR-71 was pretty much the only way to do it.
Of course, the Concorde - as well as its Soviet counterpart, the Tu-144 - routinely carried passengers at supersonic speeds when they were operational, but neither came anywhere remotely close to Mach 3.5. Both were capable of flying a bit over Mach 2, but both have been retired for many years now and there are currently no supersonic aircraft in scheduled passenger service.
Currently, exceeding Mach 1 as a tourist is pretty much limited to operations that sell rides in old jet fighters. For example, in Novgorod, Russia, tourists can pay for supersonic flights in a MiG-29 for 'only' 12,500 EUR. Not exactly cheap, but much cheaper than the $250k+ USD price tags on the proposed suborbital flights that aren't even operational yet. However, this still won't get you anywhere close to Mach 3.5. According to their website, flights will be supersonic, but will be between Mach 1 and Mach 2. The maximum speed for the MiG-29 is around Mach 2.25, so you definitely won't be hitting Mach 3.5.
With no operational non-rocket-powered manned aircraft capable of exceeding Mach 3.5, nor any planned in the reasonably near future as far as I know, pretty much the only ways for a tourist to reach those speeds will be to spend a small fortune on one of the planned suborbital flights from someone like Virgin Galactic or to spend a large fortune flying to the International Space Station on something like a Soyuz, a SpaceX Crew Dragon, or a Boeing Starliner.