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What if I set up two PID loops? Both works at same time but I take output value from one and when temperature reaches set point I take output value from another? Will this make system unstable too? I can take output value from first PID loop until temperature reaches set point -1°C, then use last output vale until temperature reaches set point+1°C and then take output value from another PID loop. 


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I guess that too would be unstable. Why do you need unsymmetrical Proportional band? Is overshoot too big if you want it to be more aggressive when temperature gets too high? Maybe try parameters that give softer approach to setpoint.

I don't know what kind of process you control, but with temperature it helps to turn off integral part when output reaches 100%, there's a function block for that.

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