Can I mix new and old lithium-ion cells

When you wire an old and new cell in parallel the new cell will initially supply more of the load current. This causes it to discharge faster until both cells share the load equally. When load current is reduced the old cell will charge the new cell for a while until they equalize again. When charging a similar effect occurs.

How much the old cell's resistance increases depends on how it is used. For lowest degradation it should be maintained at ~50% charge and the temperature should be kept below 25 °C. Charge/discharge cycles should be shallow, eg. 80% to 20% rather than 100% to 0%. If these rules are followed the resistance should not increase much in 6 months to 1 year.

So long as the cell resistances are similar, the current is not too high (ie. less than the charge/discharge ratings of a single cell), and the battery is not discharged too far (eg. no less than 3.5 V), this temporary imbalance should not be a problem. However it could be a problem if you intend to draw more current from the larger battery.