Saturation on characteristic curves of BJT
In a BJT,
Forward active mode is when Emitter Base Junction (EBJ) is forward biased and the Collector based junction (CBJ) is reverese biased.
Saturation mode is when both Emitter Base Junction (EBJ) and the Collector based junction (CBJ) are forward biased.

When you plot the output characteristics ( Ic Vs VCE ) the constant looking region is when the part is in forward active mode.

Assuming that EBJ is forward biased, when you plot the output characteristics IC Vs VCB,  realize that the VCB(sat) is -0.5V for a typical BJT ....which would mean that the CBJ 
is also forward biased at this point.The difference between VCB and VCE is almost a constant VBE ( if EBJ is already forward biased ).
Then, the output characteristics of Ic Vs VCE is merely a shift of the curves in the positive direction by a factor of VBE=0.7V

Hence VCE(sat) = VCB(sat) + VBE =  -0.5V + 0.7V = 0.2V 

Therefore when VCE is less than 0.2V ( 0.3V in your case ), the transistor will be operating in saturation region.

Points to remember: The biasing voltages are applied across Emiter-base and Collector-base