NFC - design of inductor/"antenna" made of PCB traces
Also I wonder if I do that (antenna drawn on both sides) if I could
  reach more working distance (between the reader circuit and the tags)
  than doing coil only on top side.


and


  If I need to continue the coil on bottom side also, do you have any
  recommendation about how to draw a good PCB inductor?


For near-field magnetic coupling it would be better if the coil was split between top and bottom copper layers with half the turns on the top layer and half on the bottom stacked above each other. That will allow you to use slightly fewer turns overall and get a slightly better coil Q factor. That will increase range slightly and, because coils are top and bottom you would get equal range performance from either side.

Designing coils to couple magnetism some distance requires the best amount of mutual coupling you can get between the windings of the transmit coil. Designs that are spiral are not the best but, given that you might have only one PCB layer to work with then it has to be a spiral but take note; the lower radius windings of the spiral are largely superfluous to the energy transmitted to the receiver and so, if you had (say) four layers, the optimum would be to put stacked turns on each of the four layers (just like a regular electromagnet).

So, I would look for a calculator that allows you to design the right inductance using 2 or more layers of the PCB.