Sending fax to a combined phone and fax machine

I have tried Efax.com recently, but had to cancel because I couldn't send a fax to a combined phone and fax machine.

Because we send faxes to Thailand and Indonesia, our receivers often have one single line for phones and fax.

When we manually send a fax, someone might pick up the phone and we need to ask them to press the fax button; or, the receiver will have an answer machine.

Well, if there is someone picking up, or if there is an answer machine, the fax cannot go through with efax.com. Too bad!

Is it like this with all internet fax company, or only with efax.com ?