An old service man's trick is to turn off the power (unless you like high voltage shocks), put the lamp into the spring end. Then take a flat-head screwdriver and bring your bulb in as close as you can get. Use the screwdriver to wedge the bulb into place. The socket trap covers will give a little. I run into this from time to time when working on old
signs. Have an extra lamp (or three) on hand just in case. After a few tries it becomes second nature.
Or if worst comes to worst, remove the socket trap cover (trim the length about 1/2-3/4" or even more) and slide it up a bit so that the lamp will fit on an angle. The lamps will be on a slight angle but should still properly illuminate the
sign. Make sure to fill the gap you left in the cover. I've seen
signs where the bulbs were on a 20 degree angle because the maker didn't know to oversize the cabinet in the direction of the bulbs. For an extrusion the rule of thumb is 1" longer in the direction of the lamps. I'll bet your
sign is less than 97" wide.