Many thanks to everyone that replied and gave advice and instruction. 

Here is what I've done.  I cut the original (20ga?) wire that connects the power supply to the track power clip and put electrical connectors on the ends I cut.  I purchased a simple power distribution block from Radio Shack (allows for 4 additional power feeds from the single power input) and screwed it down, out of sight and close to my power supply.  I then connected the input power wire (the PS side of the wire I cut earlier) from the power supply to the power distribution block and ran the first power feed (using the track side of the wire I cut earlier) to the original track power clip.  I then ran an additional power feed (16ga wire) from the power distribution block to second track power clip halfway around my track run (being careful to test polarity).  And Viola, I'm back in business!

Hope my description wasn't to confusing, but in short, it was clean and easy to accomplish.  I didn't change the power supply.  And using the power distribution block, some electrical connectors and the additional track power clip made the job possible without having to soldier anything.  I used wire clips from Lowes to neatly trace the second power feed wires around the track where it won't interfer with the train's operation and it can't been seen.  The single additional power feed seems to be all that was required with my run and the train is working smoothly.

Again, thanks to everyone for the help!

Texas

Once again, great input.  I'm getting closer to understanding and making an effort to correct the problem.

My power supply is a standard Bachman kit power supply.  It's tan, with a speed dial and direction control on the face.  At the top, side are two RCA style input (female) jacks.  One is where the transformer plugs into and the other is where the output wires (about 20ga) plug into.  Those output wires end with a slip-on connector (a spade) at the end that connect into a track clip that spans the tracks and makes the connection to the rails.

I appreciate the advice not to open the the power supply.  Based on the input, and the "keep it simple" advice, I think my best choice is to get a multi-point terminal strip, splice the primary power output wires to it, reconnect the primary power track clip from there, and create a couple of additional power feeds from the multi-point terminal strip that connect at the quarter and halfway point around the run as well.

I'm beginning to layout the wire runs in my mind and will proceed slowly to do a clean and efficent job.

Now, I think I have what I need to begin and many thanks for all the advice!  I have the children this weekend and a vacation planned, but I will let you know how it all turned out once I get to it.

Thanks so much!

Texas

Thanks to all for the insight and suggestions!

I think I clearly get the concept of the additional power feed(s) from a single power supply.  I also understand the need to keep polarity correct.  And I see that I can connect the additional power feed "tag" ends (other end from the power supply) to the track by either soldering or using another stock track power clip (I have an extra one).

However, I am still a little fuzzy on where/how to connect the additional power feeds to the power supply?   I appreciate Jerry's statement about connecting the additional power feeds directly to the power supply... but how does one do that? 

I could solder additional power feeds to the "ears" of the primary track power clip... I could splice into the wires coming from the power supply to the power clip to add the additional power feed wires... or I could open the power supply and solder additional power feeder wire(s) directly inside (i.e. more RC jack electrical outlets on the power supply)?  What to do, what to do?

Any pictures of additional power feeds connected to a stock power supply would be appreciated or just more detailed information on to do this easily and correctly.  I just can't picture the best way in my mind from what I have heard thus far.

I do appreciate everyones time and insight!  I'm getting closer to acting, but I don't want to experiment adding the additional power feeds to the power supply without some detailed information or a photo to look at and plan from first.

Thanks again,

Texas

Many thanks to both of you.  I now have several things to consider.

I am running Bachman steel track.  All of it I purchased new, but I did have to reduce the size of two sections to adjust the loop size to make it fit the layout.  This was fairly easy to do and the job came out clean.

I will consider brass track... but I would rather not have to replace all the track now.  Instead, I think I will try the feeder wires (I'll be careful about polarity) first.  And if that doesn't work, I will try replacing my power supply with a 5 amp model.

If I understand the feeder wire suggestion, I would solder a 10 to 12 guage wire to each of the '"ears" on the track power connector where they come in contact with the track.  Then run the wire around the loop and solder the other end to a spot on the track about halfway around (watching that the wires remain connected to the same track side).  Is that correct?

Thanks again!

Texas

I have a G-Scale White Pass Steam Engine and cars.  We have set it up on a wall shelf that encircles a 12’x12’ inside room.  We purchased the train set this year and it generally runs well in this set-up.  However, the train requires that I turn the power knob to approximately 65 for the train to begin to run and keep a fairly steady speed.  The train slows down at the far end (furthest away from where the power supply connects to the track) and picks up speed as it gets closer to the power supply feed.

It was suggested to me by a friend that I could attach a second power supply to the track on the other end to provide more even power to the train around the entire track run.  My question is to determine if this is a safe configuration (i.e. two power supplies attached to the tracks and providing power to the train simultaneously)?  Will it improve performance?  Will it negatively effect the train?

I have followed all instructions in setting up the track and train.  I have good, tight, smooth connections between all sections of the track, have used the track clips and I have the track tacked down to hold it tightly together (short nails through the holes in the plastic tie-rods provided in the track sections) as well..  I have the Bachman train maintenance kit (contact oil and grease) and have used approved track cleaning fluid.

Your advice would be appreciated.