Old Mamiya M645 lens hoods in E77 thread are great for reducing reflections, and fit most "professional" autofocus lenses, at least, those made prior to Canon (and others) moving to 82 and 86mm fronts starting with the 16-35 f/2.8L II.
Push the end of the hood up against the glass to block out the majority of your reflections.
The No. 1 model hood for 28-70mm lenses (retracted for 28mm, extended for focal lengths beyond 35mm; it will vignette even retracted at 24mm). The No. 2 model hood works perfectly for 70-200 f/2.8 lenses and 85mm without limitations, and the No. 3 model is perfect for 300mm f/4 lenses.
I believe the No. 3 should work with the 200mm f/2.8L and 180mm f/2.8D lenses, but I never confirmed this through testing as I've never seen anyone using them.
You can use step-up rings to adapt them for smaller lenses, but I don't have a lot of experience doing that.
The old Mamiya hoods are extremely durable yet pliable and can often be found in expensively in the used sections of local camera dealers, though there was a run on them when the NBA and some NCAA arenas started requiring rubber lens hoods in an effort to reduce player injury when they landed on photographers or remote cameras. I think I single-handedly made the things a "best seller" on KEH.com.
The No.1 lens hoods actually also work with minimal vignetting on 16-35/17-35/17-40 type lenses, but I don't remember when the vignetting stopped... 20mm? We usually just left the hoods retracted with the hood visible in the corners of image to maximize the image area captured by a floor remote camera.
All of this vignetting talk is related to 135 format/35mm film and "full frame" sensors that share the 24x36 dimensions. Your experience will be different for other formats.
I digress, I digress.
One can get really sophisticated and make a gobo with pieces of black wrap/cinefoil, gaffer tape, and black vinyl tape (the adhesive side of gaffer tape will reflect into your photo when adhered to glass), but zoos and aquariums don't like it when you do this during public visiting hours.