What is the best way to go about researching one's illegitimate ancestors?

Question:

What is the best way to go about researching one's illegitimate ancestors?

What is the best way to go about researching one's illegitimate ancestors? I'm not looking for links to Familysearch.org or vague replies about genealogy in general. I'm looking for specific advice on researching illegitimate ancestors.

I'm descended from Thomas Bridgwater, b.c. 1823, baptized on the 14th of January 1826 in Stoke, St. Milborough, Shropshire, England. His mother, Hannah Bridgwater (also spelled Bridgwaters), was herself christened on the 13th of August 1806 in Stoke Saint Milborough, the daughter of Edward Bridgwaters (c. 1770-1837) and Elizabeth Edwards (c. 1769-1815). She was one of at least five children: Richard (c. 1803-1833), Martha (c. 1804-1816), Hannah (b. 1806), James (c. 1808-1825) and Richard (b. 1808). Hannah had at least one other bastard son in Stoke, named William, who was christened in 1828.

Could someone please give me some clear advice on what steps I should take in order to discover the identity of the father of Thomas Bridgwater?

Additional Details

6 months ago
One person said there is no way to know, and asked how many times I'm going to ask this question. First of all, it is possible to find this information. Secondly, this is the first time that I have asked for advice on finding illegitimate ancestors in general. I'm asking about "how" I should go about it. My previous questions were similar, but they are asking for different information.

Best Answer - Chosen by Asker: 6 months ago
One person said there is no way to know, and asked how many times I'm going to ask this question. First of all, it is possible to find this information. Secondly, this is the first time that I have asked for advice on finding illegitimate ancestors in general. I'm asking about "how" I should go about it. My previous questions were similar, but they are asking for different information.