you
will need to replace that lampholder, and preferably the flex as well. It would be interesting to know the rating of the old bulb, and if it was an ordinary one or something special. The heat might have been concentrated by the shape of the shade.
If you fit an energy-saving lamp (CFL) it will generate far less heat, as well as saving you money and lasting much longer.
If the old lamp was a 100W, try to get a 20W CFL at least. 22W will be slightly brighter.
The 12W CFLs are only equivalent to a 60w bulb so OK for a table lamp, but not to light a room.
You can get CFLs with an opal globe which looks better and causes less glare if there is no shade. The stick lamps and spirals are cheap and bright but rather ugly if they are not concealed by the shade. The frosted globe make a slight reduction in brightness.
Some brands are slow to come bright, the latest Osram are notably quick.
From time to time, CFLs are subsidised as part of energy-saving campaigns, and it is worth buying your spares when that happens. But try one first in case (like some Philips) it is slow to brighten or you don't like the colour. "Warm White" 2700k is quite a nice tone.