Increasing the bandwidth can be done by increasing the substrate height and/or using multiple radiating or non-radiating resonators.
The first method is relative straight-forward and will give you result with limited amount of iterations. The second method (multiple resonances) requires good skills as creating other resonance (with slots or other patches), mostly affects the first resonance also.
I would recommend you to read a good reference on microstrip antennas, for example: "Compact & Broadband Microstrip Antennas", Kin-Lu wong.
Using a second non-radiating resonator will give you an increase of useful bandwidth (VSWR=2) of almost factor 2.5.
Note that you can increase the BW by introducing loss, but this also reduces the radiation efficiency (hence gain) and that is mostly not desired. Other option is to lower eps.r, but this will result in a larger patch size.
Increasing the bandwidth can be done by increasing the substrate height and/or using multiple radiating or non-radiating resonators.
The first method is relative straight-forward and will give you result with limited amount of iterations. The second method (multiple resonances) required good skills.
I would recommend you to read a good reference on microstrip antennas, for example: "Compact & Broadband Microstrip Antennas", Kin-Lu wong.
Using a second non-radiating resonator will give you an increase of useful bandwidth (VSWR=2) of almost factor 2.5. In your attachment I see you use a parasitic patch above the fed patch. With careful design this should create more bandwidth. I would expect a more or less inversed molar shaped SWR or S11 pattern.
You have already lots of air under the metallic patch, so reducing effective Eps.r is no option.