Wendy..not to worry..This should be an easy problem to solve.
You are correct, in all likelihood your lift linkage has come loose and the linkage has fallen down the overflow pipe. The point in the tee where the stopper actually stops the drain is a tapered fit so the stopper and linkage will not normally fall below that point, but even if it did, it will still stop at the trap so we can recover it fairly easy.
Lets try the easy way first. Remove the overflow cover on the inside of the tub then get a piece of stiff wire such as an old coat hanger about 2.5 to 3 feet long and bend a small hook on the end. Insert the wire down the overflow pipe hook end first and see if you can snag the linkage and lift it up to where you can grab it.
If you can't recover the stopper assembly with the wire hook you will need to dissassemble the overflow standpipe. Some overflow heads are mounted with the same screws that hold the overflow cover in place and others have a separate hanger behind the overflow cover. If you have the separate hanger you will need to remove the two mount screws and the hanger bracket.
Next go in through your access cover and examine the pipes. You will see a TEE where the tub drain meets the overflow riser. There should be a compression fitting directly above the TEE. Unscrew the compression nut and the overflow riser will lift up and out of the tee. That should expose the linkage and your missing stopper. Lift the linkage and the stopper up and out and lay it aside. Now reconnect the overflow riser but do not tighten the compression nut at this time.
Realign the overflow head with the hole in the tub and tighten the compression nut slightly to hold it in place. ( I like to put a light film of clear silicone sealer on the rubber gasket but that is not necessary if you dont have it).
Examine you stopper assembly and you will see there is two peices of wire and a 4" to 6" section of flat brass metal formed into a U shape at the top end. On the top end of the linkage the brass U shaped piece has a hole drilled through both sides. Now examine the lever that is attached to the overflow cover and you will see that the inside end of the lever has a hole drilled through it from side to side. The lift arm is placed between the two parts of the U-shaped linkage and a cotter pin is put through the linkage and lift arm to hold it in place. The odds of finding the original cotter pin are slim to none, and even if you found it, it would most likely be too rusty to re use. Fortunately you can buy a replacement brass cotter key at any hardware for about $.20. ( You could also use a short piece of 12 guage copper electrical wire to make a new pin. Just put the wire through the hole and bend each end at right angles so it can't slip out.
Once you have the linkage reassembled, slide the stopper down the overflow pipe and screw the overflow cover in place, then tighten the compression nut on the overflow riser connection and close up the access hole and your done.
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