I have an existing sub-panel fed by 60 amp, 220 breaker on main panel. I need to add 20A, 220v but the sub-panel does NOT have space for a double breaker. I want to add an additional sub-panel immediately adjacent to this one. Can I double lug from the existing panel to the new panel?

Attached is a photo showing proposed wires (blue Lines) to new panel.
Notes:
1. Each 240v breaker supplies 3 ea 1500 watt Halogen Flood lights
2. One of 120v breakers supplies an outlet used for leaf blowers, and small, dorm type refridgerator.
3. Other 120v breaker supplies a switch not currently used.
4. Need 240v, 15 amp breaker to supply small mini-split A/C system