I'm going to describe two approaches: 1. DNA Testing. 2. Paper/Electronic Brute-Force.
Approach 1: DNA
The first thing that comes to my mind is DNA Testing. If your test returns someone who is:
1) a fourth, third, second, or even first cousin;
2) not related to your mother;
you should be able to track backwards to some degree. How close you get to your father depends on how close of a match is found. Many providers give possibilities to send messages to genetic matches. This gives you a good opportunity to talk to those who are almost certainly related to you on your father's side. It does require that you know something about your mother's family line if you want to rule out those matches which come from a common ancestor with your mother (unless you choose the Y-Chromosome Test.)
It is something of a roll of the dice; someone with whom your father shares a common ancestor must have taken the test. I was born in Utah, home of the Mormon Genealogical Phenomenon. Many people around here have taken tests from various vendors, so many of my close relatives have come up as matches. My wife, on the other hand, grew up in the Soviet Union and even started her career there. I met her when she came to the US to advance her studies. She has few matches, and most are 4th-cousins or more distant. Few people from Russia and other former parts of the Soviet Union have taken a test (with the exception of Central Europe - the Baltics, Poland, Czechia, Slovakia, etc.)
In making your decision as to which test to take, I suggest having a look at the following book:
BETTINGER, Blaine, The Family Tree Guide to DNA Testing and Genetic Genealogy, Family Tree Books, 2016
I got my digital version from Google Books. That option is no longer available. There are digital versions available for as low as $2.99 (including the necessary e-book reader) at
Amazon (and free Kindle App), B&N (and free NOOK App), and the publisher.
You can also look in WorldCat for a paper version near you, or check with your favorite paper-style bookstore.
The author of the book, also the author of The Genetic Genealogist blog, included this part of the book in a blog post:
One reviewer very concisely sums up my feelings about the book as Informative. Understandable. Solid.
If you don't feel like that book, I'll link some sites that give reviews and explanations of different DNA tests. A Y-Chromosome test might be best for you, since it is your father's information you're hunting, but that doesn't mean it will be the best test.
Informational Links: mashable smarterhobby Forbes thoughtco thegeneticgenealogist (archives)
Approach 2: Paper and Electronic Research, a.k.a. Brute Force
You're of an age where you might be able to do some online-search magic without asking for younger peoples' help.
The strategy here is to basically do searches using combinations of the following, with most important coming first: the surname/last name you know, a place (if available), the profession you know.
I ran an example to show you what you might deal with. I searched for one of my grandfathers who would be over a century old if he were alive today. He has a common first and last name, and lived for significant time periods in two small towns and several medium-sized towns.
Using his full first name and his full last name, as well as the two states where he spent the majority of his life, it took me 15 minutes to find him - but I knew exactly what I was looking for. The find came through entering the information into a major search site and then inspecting sites such as whitepages, beenverified, fastpeoplesearch peoplefinder, etc. Be VERY careful that you don't end up paying for something accidentally.
When I went down to a first initial, last name, and states, it still took about 15 minutes, but that was helped by adding a small town to the search, then looking at the previous search results to make sure that the "find" was actually there. Note that I didn't use the first initial until I found results for the last name and place. This was, once again, found through the directory/people-finder method.
With first initial, last name, states, and occupation, it took a while. I accidentally stumbled over a record that wasn't there because of a link between Grandpa and his occupation. I don't know if I would have found it if I didn't know more details. The find was in a yearbook where Grandpa was mentioned in passing.
The next step has been more useful for me as I've helped others find information about relatives - Use Social Media. This will be especially helpful given that your father has an uncommon last name. However, I'll show the type of results that can be expected. From a co-worker's Facebook (someone with whom I don't share a lot of friends,) I put in the last name and each of the two states where Grandpa mostly lived. The test was to find out how hard it is to find people whom I know and who could tell me about Grandpa. Of course, it's not a perfect test, because I'm going in knowing which people I'm looking for.
Here, I'll list the ranges of "hits", i.e. the search result number(s) where there were people who I know could give me information about Grandpa if I didn't have any. Think of it as the amount of people you'd need to contact on social media before finding one who might give you information.
State 1 State 2
41-50: 2 hits
51-60: 1 hit ... ...
... ... ... ...
110-120: 1 hit
120-130: 1 hit
So you don't give up hope, I'll also include two searches for uncommon names that I've used in Central-Eastern European searches for my wife's relatives. This is all using Facebook or one of the out-of-the-US social media sites that I'll discuss further down.
Very Uncommon Name, Known to have moved to the US
Facebook (only counting Facebook
people with US stuff) (all results)
1-10: 4 1-10: 0
Somewhat Uncommon Name, Not in the US
Facebook International Site
0-10: 1 0-10: 1
11-20: 2 11-20: 1
21-30: 1 21-30: 0
31-40: 4 31-40: 3
41-50: 1 41-50: 2
International Social Media Sites
I've found these very useful in helping others find relatives if such people are first- or second- generational Americans. They have also helped me in finding relatives of Russians. I'll put in some links to lists of these sites, then I'll do a brief list of my own. Sometimes links go dead, after all.
There's also some interesting information about social-media usage per country and with use-by-age statistics at https://jacobsmedia.com/social-media-globally/ I'll include one image from this site.
International Social Media Sites
Russia (obviously comes with risks)
ВКонтакте (VKontakte, vk)
Одноклассники (Odnoklassniki, ok)
Note also Yandex, Viber, Telegraph, ...
China (obviously comes with risks)
Note also WeChat, QQ, Baidu Tieba, ...
mixi (requires validation of a Japanese cell-phone number)
Note also KaKaoTalk
Netlog (world-wide users, notably those in parts of the Middle East and Asia)
Orkut (lots from Brazil - note that it's prone to viruses!)
Taringa! (based in Argentina)
See the list here.
These all go along with Facebook, Google+, LinkedIn, Twitter, Instagram, Reddit, WhatsApp, ..., which are used throughout most of the world.
It's a hard problem, as many genealogy problems are. You'll have a better chance of success if you can use DNA techniques. However, I know that there are plenty of reasons people choose not to use this route. I hope the other search methods described here will help you if you're one of the people who doesn't want to use DNA.