Your second (modified) circuit should work. The output stage and the on-off ratios could be improved for efficiency, but it should easily work at 1m even at a much lower drive level, that is, at a lower LED current. Calculations using your component values yield a nominal burst length of about 26 cycles ON and 52 cycles OFF, which should be OK for a TSOP1738.
Here are some thoughts:
1. Is there a strong ambient light shining on the receiver module? Other light sources can reduce the sensitivity of the receiver. It should be shielded in directions other than the front. A guide tube at the front will also help.
2. Judging by your resistor values, I assume you're using 1% types. What about the timing capacitor? If you're using a ceramic disc capacitor, they have wide tolerances and could place the frequency well of the ideal 38kHz. Do you have any way of checking the frequency?
3. As someone else pointed out, the LED's wavelength should be 940 or 950nm. If the ones you're using happen to be, say 850nm types, that will greatly reduce the sensitivity too. If you don't know, can you get known ones? For example, the Vishay TSAL6200 is a good one.
I've been working a lot with IR transmissions for some time now. With about the same LED current as yours (~100mA peak), I get reliable reception at 4-5m using no-name IR LEDs from the local market, and up to 15m using Vishay TSAL6100 LEDs. The 6100 is more sharply focused than the 6200 and has a stronger axial intensity, but has a narrow angle of emission and is best suited for fixed point-to-point communication.