Use 555 Timer Voltage Doubler to Increase Range of Transmission
That circuit should work, and will provide the necessary current at your doubled input voltage.  Currently at 9V, it looks like your transmitter there will draw a bit over 9mA fully loaded.  At 18V, it would draw 18mA.  That extra 9V and 9mA doesn't come for free though, you will have 1/4 the battery life that you do at 9V.

1N4001 are rated for 50V reverse voltage, 1N4004's are rated for 400V reverse voltage, so for that doubler, there shouldn't be much difference.