Bingowings said:

If you have just had the first one blown up wouldn't it make more sense to build lots of small Death Stars in different locations (each which could be finished more quickly) than an even bigger one that takes ages to finish and can be blown up in one go?

Who says it was started only after the first was destroyed? They could have started building it years before Yavin.

You see... in ww1 era if they sunk a battleship then the country made a bigger one. They didn't make a few smaller ones instead. Same in ww2 with aircraft carriers (except Japan at the end of the war). So it makes perfect sense.