I estimate that half the slant needle Singers I find in the wild have a broken upper hinge on the nose door. This is especially true of the 500 series (500a & 503a) but I have also seen it on 401's and 600's. Many of these doors are held on with clear packing tape and that's a band-aid fix but I prefer to have my machines LOOK nice, even if they have been repaired.
So, here is the nose door from the 503 I am rebuilding. It has the common feature of half the upper hinge broken off and missing, so it cannot just be glued back. With the broken piece went the hinge pin, so we also have to find a replacement pin. I normally use a finish nail, but this is the shaft out of a pop rivet.
I mix up some epoxy and fill what's left of the channel for the pin.
Next, I position the replacement pin so that it is at the correct angle and press it into the epoxy. It is best to balance the pin in the slot so that it does not slump down before the epoxy hardens. Then, I slather epoxy over the pin to help hold it in place.
24 hours later, I slice off the excess epoxy and excess pin from the top of the hinge. I use a rotary tool with a cutoff wheel. If I applied too much epoxy, now is the time to grind or file it down so it doesn't interfere with the opening or closing of the door.
The original pin is about 1/4" long, but you will find it easier to install the door if you leave the replacement pin a bit longer. With both upper an lower pins the same length, you have to align both with their respective holes to install the door. If you leave the upper pin longer, you can insert the upper pin and the extra length holds the upper hinge in place while you fiddle with the lower hinge pin.
TA-DA! the 503 now has a functioning nose door.