Just ask for a lawyer. Don't worry about having to sit there in awkward silence for hours if it comes to that. There's no law that says if you can't find a lawyer in X amount of time after asking, you don't get one. You don't have to schedule a time for a lawyer to come in to get you.
Once you ask for a lawyer (which should effectively terminate the questioning, or at least it should), one of three things may happen:
(1) you will be arrested and charged. You gave them no new evidence to use against you, so you're in the best possible position.
(2) you will be "detained" for some amount of time while they go about sketchy, quasi-legal moves to gather more evidence against you (applying for search warrants for saliva/blood sample, your vehicle, home, etc. and they want to keep an eye on you until these are approved and executed). Depending on how their fishing expedition goes, either #1 or #3 will follow.
(3) they will release you.
Regardless of what happens, DO NOT TALK TO THEM.
The one thing that no one ever mentions is that the police usually don't follow the rules, and the courts usually don't penalize them for it. The only way to protect yourself is to shut up. Don't even make small talk. It's a chess game and there's no move you can make where you don't lose. Do and say nothing.
You know exactly what your lawyer is going to do? He's going to walk into $LOCATION, announce "I am the attorney for John Smith. Take me to him immediately.", greet you, then say "My client refuses to answer any further questions. Is he under arrest? If not, he is leaving. Here's my card if you have any further questions about this matter."
And that will, most likely, be your entire engagement with this attorney. If you need one in the near future, you can hire one after a more informed search, and have your existing attorney brief him on the procedural history -- to whit, that you were briefly detained and that he did exactly what every other attorney in the phone book would have done under those circumstances.
P.S. In the United States, if you're under arrest, you are entitled to a public defender. Heck, you can even say "I won't speak to you except under advise of counsel. I don't have a lawyer at present, so until I find one, I'll accept the services of a public defender." If they refuse a public defender, you're not under arrest, and if you're not under arrest, you should probably leave.
Is that actually how it works in the US? A judge in Canada recently lamented this as a "misconception brought on by watching too many Hollywood movies", and pointed out that (at least in Canada) you can make as many phone calls as you need in order to get a lawyer.
You're not obligated to talk without a lawyer. The police won't give you what you need to find a lawyer? Fine, just don't talk. If they're stupid enough to say, "you wasted your one phone call, no lawyer for you" then just sit there.
The legal system is run by humans and doesn't generally have silly gotchas like these. There's no such thing as, "you couldn't find a lawyer in ten minutes with one phone call, so you don't get to have one, gotcha!"