Different tuners have different impedance ranges that they can cope with. The quickest clue to this is the shortest wire that they can tune at some low frequency - some require 10m of wire at 80m, some will tune 5m. There's not much you can do about the tuner range.
However, the tuner is not completely democratic about all forms of high SWR - some are easier to tune than others. It's possible that if you shift the impedance from one side of the chart to the other, it might have a better chance.
You can alter the impedance seen by the tuner by adding (or removing) some coax. If the SWR isn't too high, this might move the impedance into a range that the tuner can cope with.
Try 1/8, 1/4 or 3/8 wavelengths. Half a wavelength is too much, the impedance is back where it started. Remember the velocity factor of your coax, which is 0.66 for solid PE.
For 80m you could start with 5 or 6 m of coax, and see if it helps.
At 10m the same piece of coax will be at least a full wavelength, but probably a bit more or less, so it is likely to shake up the impedance there too.
Moving the tuner closer to the antenna has the same effect, but has the added benefit of reducing the amount of high-swr coax, which will reduce the losses. Ideally the tuner would be right at the antenna, after the balun. Some tuners have a remote control for this purpose, a small box with the same buttons and LEDs. You can easily build this box.