Again, I would check for codes in the HVAC unit.

Next, are you having to add coolant? If the level is low, there will be air in the system which will cause strange temps from the heater.

 

If your 01 accord was like my 95, then the temp control **** mechanically adjusts the antifreeze valve. Sometimes the **** would crack along the axis and turning the **** barely adjusts the temperature.

See if the arm on that control valve moves through the whole range when you turn the temp control **** from cold to hot.

 

1. The HVAC trouble code retrieval instructions in the Common DIY and Other Info. thread is applicable to the 7th generation Accords (2003-2007). The 6th generation Accords are instructions are slightly different. Since you have Climate Control (automatic and semi-automatic), these are the instructions for climate control trouble code retrieval (not manual A/C) (see attached images).

Climate Control
General Troubleshooting Information
How to Retrieve a DTC
The climate control unit has a self-diagnosis function.
Running the Self-diagnosis Function
1. Turn the ignition switch ON (II).
2. Set the temperature control dial on Max Cool, then slowly to Max Hot.
3. After 1 minute, press the AUTO button and then the OFF button. Continue to hold both buttons down. If there is any abnormality in the system when both buttons are pressed, the temperature indicator will light up the segment (A to N) corresponding to the error. The temperature indicator will then alternate every second between displaying "88" (all segments lit) and the error segment. If there is no abnormality, the segments will not light up.

Resetting the Self-diagnosis Function
Turning the ignition switch OFF will cancel the self-diagnosis function. After completing repair work, run the self-diagnosis function again to make sure that there are no other malfunctions.
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2. If there is a coolant leakage somewhere, they would be a concern as it could possibly affect the heating/cooling issue.

3. The climate control system automatically selects the proper mix of conditioned and/or heated air that will, as quickly as possible, raise or lower the interior temperature from its current level to the set temperature. It uses the sunlight sensor, an outside temperature sensor, an inside temperature sensor, and an evaporator temperature sensor for input to control the blower controls, air delivery, and A/C compressor controls either automatically or semi-automatically.

When you set the temperature to its lower limit (60F/18C) or its upper limit (90F/32C), the system runs at full cooling or heating only. It does not regulate the interior temperature. When the temperature is set between the lower and upper limits, the system regulates the interior temperature to the set value.

4. The issue you're describing happens when the temperature is set below the upper limit (i.e., 90F/32C).

This may indicate an issue with the climate control unit, temperature sensors, or air-mix motor control circuit or linkage to the doors.

5. As previously mentioned, the climate control system can operate automatically or semi-automatically.

Semi-automatic Operation
You can manually select various functions of the climate control system when it is in FULL AUTO. All other features remain automatically controlled.
Making a manual selection causes the word FULL to go off.

However, when you select A/C OFF, the system cannot regulate the inside temperature if you set the dial lower than the temperature outside.

Try selecting A/C OFF, to see if you can now control the heat. If you have a thermometer, compare that with the temperature showing on the climate control display.