All Elgins from the teens forward were sold via Sears. With a pic or a serial number, we can probably determine which manufacturer produced the frame for Sears. (At least 3 different companies made Elgin-badged bikes for Sears.) Some folks, armed with the right Sears catalogs, might be able to ID with just the model number, but i'm not one of those ppl...
***Edited for clarity-- there were very early manufacturers producing bikes in the 1890s and 1900s in Elgin, Illinois, some of which were sold as Elgin.