Do you know if this result is repeatable ? You only shown one device maybe you should try to repeat this experiment if that is not already the case.
Also, the trend from 1-3 cycles is increasing, does it recover to the 340 mAh.g-1 value after further cycles?
What is the duration of the cycles? What is the discharge rate? One could imagine that very high discharge rates could cause some sort of surface reconfiguration on initial discharge cycle.
This behavior is typically observed when an electrode is not fully wet in the initial cycles. Theoretically, it does not make sense that your graphite anode shows an increase in capacity over cycles since Li ions are consumed more and more. I would suggest examining the amount of the actual electrolyte impregnation. The cause can be vary, e.g. coin cell could compress electrode too much etc.
I agree with Jung Hoon Chae . I once used less electrolyte solution when making a batch of coin cells, and while most of them wouldn't pass any current, one of them increased capacity until the expect value was reached. When I added more electrolyte solution to the next batch of coin cells they all behaved normally.