Subdomain in .htaccess file only works with index - .htaccess

Question

I have a subdomain setup in my .htaccess, which only seems to work with the default index.html page. I'd LIKE it to work for ANY page in the folder corresponding to the subdomain. Edited for privacy, assume my domain is example.org. The pertinent parts of the file look like this...
#subdomain
RewriteCond %{HTTP_HOST} ^subname\.example\.org$ [OR]
RewriteCond %{HTTP_HOST} ^www\.subname\.example\.org$
# (a few lines added by my hosting company deleted -- see below)
RewriteRule ^/?$ "http\:\/\/example\.org\/subname\/" [R=301,L]
So the result of the above is that if I have an index.html page in my 'public-html' (root?), http://example.org and a different index.html stored in a sub-folder (having the same name as the subdomain), I will get this expected result, which works...
browse to: http://example.org results in viewing http:// example.org/index.html
browse to: http://subname.example.org results in viewing http:// example.org/subname/index.html
Great so far. This is what I expected when I created the domain name. However, given a specific file myfile.html stored in the subname folder, I would expect this to work also, and it doesn't...
browse to: http://subname.example.org/myfile.html results in a 404 error.
This despite the fact that browsing to http://example.org/subname/myfile.html works fine. In that case myfile.html is displayed. So is there anything I can do to modify the subdomain code to get the result I'm looking for? Namely, browsing to http://subname.example.org/ANYFILE should work as well as browsing to http://example.org/subname/ANYFILE, regardless of what 'ANYFILE' is. This, after all, is one of the main reasons I set up the subdomain to begin with!
Note: I confess that I relied on my hosting company's cPanel utility to create the subdomain code, so I asked for their tech support for help first. Long story short they didn't. Maybe what I hoped for is not actually possible?
Also, the lines I deleted' from the code had to do with something called "well-known/acme-challenge", added by my hosting company at some point. Since removing them had no effect on the behavior I've described, I left it out to avoid clouding the issue.

Answer 1

RewriteRule ^/?$ "http\:\/\/example\.org\/subname\/" [R=301,L]
This only "redirects" the document root. To redirect all URLs you need to change the above to read something like:
RewriteRule (.*) http://example.org/subname/$1 [R=301,L]
The $1 backreference refers to the URL-path captured in the RewriteRule pattern, ie. (.*).
No need to backslash-escape the colons, slashes and dots in the substitution string (that's typical of cPanel).
Also, the lines I deleted' from the code had to do with something called "well-known/acme-challenge", added by my hosting company at some point.
Those lines will likely be required when the (Let's Encrypt?) SSL cert auto-renews. (Although the above redirects to "http" - are you not using HTTPS?)
UPDATE:
RewriteCond %{HTTP_HOST} ^subname\.example\.org$ [OR]
RewriteCond %{HTTP_HOST} ^www\.subname\.example\.org$
Just as an aside, these two conditions could be reduced to a single condition if you wanted. For example, the above is equivalent to:
RewriteCond %{HTTP_HOST} ^(www\.)?subname\.example\.org$