Every thing that you can do to help document the history of the property will help reduce the surveying bill. Call the city and/or the county and find out which department keeps the survey records. Go to that office and ask for help finding records that show any existing monuments, even for the neighbor's property. If the adjacent properties have been surveyed, then the monuments for those surveys will establish the corners of your property. Even if only one or two of the corners have already been established, you will be way ahead. Really expensive surveys happen when the nearest reference points are very far away, requiring a lot of field work.
If your research finds that monuments have been placed on the corners of your property, then go look for them with the metal detector.
The quote from the reasonably priced surveyor was in the ball park, but you can help him by doing the research. If you can walk him right to at least one monument on your property, it will make his job much easier and faster - which should translate into less expensive.
I have tromped over nearly 1,000 acres of timber land in the last 20 years looking for old monuments, and found nearly all the existing ones with a compass, fabric open reel tape measure and metal detector. Only in the last year was I using a GPS, and it made it much easier. You still need to find an existing monument to start from, even if it is on an nearby property.
By the way, you are usually much better off to find existing monuments. One reason is that there is an order of authority in establishing boundries, and "found" monuments have the highest rank.