Ok, I'll spell it out since no one else will.
there is no tip that is thin enough to allow you to solder through a non-plated through hole. not gonna happen for multiple reasons.
Making your own double sided boards is easy except you don't get plated through holes (or solder mask). You *could* make a board with PTHs but it's a lot of additional work and simply not worth the effort. When I design a DS board for home fabrication I apply what I call "Non-Plated Through Holes Design Rules". This means that you have to find places where you can make a connection between the two layers. There are two kinds of places where you can do this: vias and component leads that are accessable from both sides for soldering and thus making a connection between the two layers.
Components that give access to both sides (NPTH compatable): leaded resistors, axial electrolytic caps, leaded ceramic caps, unsocketed dips, to220s... You can terminate a top layer trace at one of these pins.
Components that are incompatable with NPTH: sockets, flush leds, dsub minis, rj11/45/..., terminals, most relays, radial caps... Do not terminate a top layer trace at one of these pins.
Vias are pretty easy to make work in NPTH boards. You just run a wire through the hole and solder both sides. some people use little grommets that have to pressed into the via hole but I think that is a huge waste of time when you can simply solder a wire.
Design a board or two with NPTH DRs and it becomes second nature. Here's a pic of one I did recently. Note the ovals - the top one is where I use a dip pin to connect top and bottom grounds, the bottom one is where I use a diode lead. The large dip on the right is socketed.
**broken link removed**
You might be able to save your board by taking some really thin wire (say one strand of stranded 16 ga wire) and thread it through the holes where you have a problem. then solder the wire to the top layer trace in such a way that it doesn't interfere with the fit of the socket. You might have to drill out the holes a little bigger. then insert the socket (or other component) and solder the bottom wire, pin and trace together. I've tried it once and it was a real pita - just use NPTH design rules and all is well.
Phil