I've always wondered how one can access the splice without playing the tape to bring the spice out of the wound tape?
If the splice has not yet broken and you don't have any tension problems with the tape (another common problem with old 8-tracks) then here's what you do.
1. Take a look at the track listing for a particular track, for instance, Track 1. Pick a track where the song actually ends at the end of the track rather than fading out and continuing on the next track.
2. Let's say you picked Track 1 of a Beatles tape and the last song was Let It Be.
3. Pop the tape in the player and play track one all the way through to the end of the Let It Be song. You are in no danger of ruining the tape until it reaches that splice point.
4. As soon as the song is over and you hear silence, immediately eject the tape from the machine.
5. You have now stopped the tape a short distance from the foil splice. Now, using your finger, manually roll the tape forward by pressing your finger against the rubber roller on the right side of the cart. Hand roll the tape until the foil splice appears. Be patient, not all tracks are the same length and several inches of tape might need to be rolled out before reaching the splice.
6. Once you see the splice, center it at the opening at the top of the cart. Now, using something like a pencil, pull out just a bit of tape so you have access to the underside of the splice as well as the foil surface.
7. Test the foil splice to see if it easily peels off. Sometimes they literally fall off, sometimes they are barely still adhering and you might need to pull it back with a fingernail, sometimes they are still adhered solidly. If it is still sticking firmly, don't peel it off.
8. If the foil splice falls off or is easily removed, cut a piece of foil splicing tape roughly the same length as the orginal piece and replace it.
9. To be extra safe I always also put a piece of audio splicing tape on the underside of the splice opposite the foil splice. That way if for some unlikely reason the splice were to come off in the future your tape would still be intact and would not wind around your capstan. You'd still need to replace the foil splice in this case because a tape will not change tracks without the splice.
10. Once you have completed your repair use the same finger rolling technique on the roller to take up the excess tape you pulled out of the cart.
Audio splicing tape is made by 3M and is sort of a milky white color. Foil splicing tape is becoming much rarer and harder to find, but you should be able to find some on Ebay or other online sources. I have not updated my old 8-track website for years, but most of the information should still be pertinent. There are instructions for all kinds of tape repairs as well as resources you can try for finding repair materials.
http://keepontrackin.bravepages.com/answers.htm
Hope that helps.