FTDI-based programmer works with ATmega but not ATtiny?
Please try with -B20 or -B50 -B100 flags if your programmer supports.
Best way to flash an ATtiny is, USBTiny build from another ATTinyX5 :)

If you run your ATTiny on some low speeds (I run mine @ 8khz for example, yeah it's possible to clock it), you need to slower SCK pulses at programmer.

Edit: I want to add this, avrdude doesn't allow USBTiny with higher flag than -B250. But Since I am experimenting with ATTiny clock (32khz, 8 khz...), I need -B400 even -B2000... So just recompiled avrdude from source and increasing limit to 9000 and have no issues. Hack works good. Also you don't need wait to flash whole program. For rescuing low hz fused attiny, compile any program with 1/8mhz flags and CPU settings. Than just flash for a second and abort. This will restore fuse settings and you can reprogram whatever you want with high speed, for example -B1 flag. (I use this setting as default since it's fastest and working with 8Mhz 16Mhz PLL clocked ATtiny85. But -B1 will not program your 1Mhz fused chips.)

Regards,
Erdem