I know this question is kind of old but there's an easier way to get the URL to the older zip files.
Using a combination of answers here, you can calculate the download URL of the zip file for the specific version you want.
Determine the version number you want to get (we'll use 3.8.7.4 as an example)
Look on the timeline (http://www.sqlite.org/src/timeline?t=release) to figure out the year in which your desired version was released (3.8.7.4 was released in 2014)
Normalize the version number into exactly 7 digits. Expand each piece into 2 digits with leading zeroes except for the initial 3 which should remain 1 digit. For example 3.8.7.4 becomes 3080704. 3.13.0 becomes 3130000. (If there is no 4th period delimited piece, use 00)
Using your normalized version number, plug it into one of these formats, depending on what you're looking for (Replace the text '7DIGITS' in the urls below with your normalized version number, replace the text YEAR with the year in which the version was released
Source: http://www.sqlite.org/YEAR/sqlite-src-7DIGITS.zip
Amalgamated: http://www.sqlite.org/YEAR/sqlite-amalgamation-7DIGITS.zip
So our example versions become
http://www.sqlite.org/2014/sqlite-src-3080704.zip and http://www.sqlite.org/2014/sqlite-amalgamation-3080704.zip
I haven't tried this for every version but my 3 test versions worked. I would imagine the other download types (like precompiled binaries, documentation, etc) work as well.