I would do a cover crop since it does add organic matter and depending on the crop you choose you could also sequestor some nitrogen.
Since the garden is less productive, and you probably did not rotate crops in the garden, the pH and the nutrients may be high, low or just out of balance with each other. I would do a soil test after your last crop, or now so you can get your results before you leave. You can add the fertilizer recommendations later, but you could make the pH correction if you had to before you leave. Changing pH takes about 6 months, so you will need a head start on that. Depending on your compost source, compost and chicken manures are very alkaline, also if you have been adding wood ashes for the potassium, or you have been liming just for BER or just because some people assume the garden only gets acidic over time and add lime without checking pH, you could end up with a high pH. P.S. I am guilty of doing the latter, I limed every two years until, after 8 years, I could not get okra to grow anymore, only cabbages. Most plants like pH between 5.5-7.5, but few plants (most desert plants) tolerate very alkaline conditions. Some plants like blueberries (pH 3 - 5) and baby's breath 7.5-8.0) like the extreme ends better.
If there isn't going to be anyone to water the garden or you don't have an automated system while you are away, you could solarize the garden instead. July and August are the best months to solarize.