Hi guys, I am thinking that my alternator may be on its way out. Here are my voltage readings, taken directly from the battery posts with a brand new multi-meter.

12.73V not running
13.30 Running
12.85 Running w headlights
12.59 Running w headlights and heater fan

I am pretty sure that it is going bad but would like a second opinion. Thank you.

 

Recently installed a new fully charged battery?

Those readings would indicate to me that your alt. is OK & battery is a fully charged. I'd not worry about it unless seeing initial voltage reading prior to start drop. Constant higher voltages while running may be the result of faulty battery needing a top-off or voltage regulator over charging battery.
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The battery is new and the alternator looks new. I just bought this 97 LR Disco. I drove it a handful of times and it wouldn't restart after driving 20 miles and stopped for about 5 minutes to go to the store. I was close to home so I walked home to get my other car to jump start it. But then it fired up without the jump.
This has happened twice.

 

have you checked your ground cables ? a poor connection can cause all kinds of stuff to happen, they are known to corrode and the connection at the frame & engine can be rusty

 

Problem before and/or after new battery? New batt and still problems could be loose terminals as suggested or alt. If no problems since new batt chill.
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those number are barely what I would consider boarder line, Iirc out put should be 13.5 or better.

 

D1 sorry

 

That is what I thought. I should mention that even though the truck restarted on those two occasions, I had to put the battery on the charger because the battery charge was low.

I will check the ground cables as mentioned above. The terminals at the post are new and clean.

 

Agree, probably would've been boarder even with the best battery. Checked my lights on bright/off today and only saw a 0.1 difference. Likely a bad alt.
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Im not convinced it's an alternator. When it wouldn't start at the store, but then started fine without the jump, what exactly do you mean it wouldn't start? Weak crank? No crank at all? starter solenoid clicked but starter didn't turn? Or no power whatsoever (no lights on dash etc.)?

Could be a bad battery connection. I had a pajero that would do this occasionally, where there would be lights on the dash but the solenoid would only click. New battery etc. The terminals were clean looking however the clamps had enough light oxidation built up that occasionally they would not make a good connection for enough current to flow to the starter. Wire brushing the inside of the clamps until the metal shined and making sure they were tightened down well fixed the problem for good.

Check your starter circuit and charging circuit for bad connections by doing a voltage drop test.

1) remove fuel pump relay or similar so that the engine won't start when cranked.
2)take volt meter and put + lead to Batt+ POST, negative lead to neg-Batt POST (not clamp, but the post.)
3) Crank engine and record voltage

4)Volt meter + lead to + Batt terminal on starter, negative lead to starter housing. Crank and record value. Values should be within 1 volt. If they are not, then there is voltage drop in the circuit.

Check positive side of circuit by placing + lead on battery + post. Negative lead on +battery terminal on starter. Crank and read the value. Total voltage drop should be less than about 0.5 V. (Make sure your volt meter is on a setting that can read this. 2V range, not 20V range.) If the voltage drop exceeds this, repeat the process for every connection in the circuit between the battery terminal and the battery terminal on the starter. Each individual connection should be less than 0.1V drop. Find the problem connection and repair.

For negative side of the starter circuit, check by putting positive volt meter lead to starter case, negative lead to negative battery post. Looking for less than 0.3V when cranking. Check voltage drop along grounding strap from battery to engine as well (less than 0.2V ideally)

That should get you by checking for a bad starter circuit. Theres a similar method for checking charging circuit voltage drops

Edit: May as well include how to check charging circuit while you have your volt meter out. Make sure to check alternator charging circuit with fan and lights turned on, and engine over an idle, around 1500 RPM

Place Positive lead of volt meter on Alternator output terminal. Negative lead on Battery positive post. Crank and check for voltage drop of more than 0.5 V. If found, isolate the connection and repair.
Place negative terminal on alternator casing, positive terminal on negative stud. Crank and check for voltage drop less than 0.2V. If the alternator isn't grounded through it's attachment bolts and has a separate ground strap, check that too.