I use a Minolta Spotmeter F which only has the 1° spot mode, in ambient light with 100 ISO goes down to EV 1 according to the producer. If you are planning a new (second-hand) purchase, I find the Minolta very good stuff, but it's inconvenient to have two different lightmeters, one for spot measuring and one for incident metering.
As Chan Tran says, your Sekonic goes down to EV 3 but that should be more than sufficient in normal situations. Maybe it's a stupid advice, but I would check battery contacts and clean them well.
Certain lightmeters have two different cells for highlights and shadows (Pentax spot meter IIRC). Maybe your lightmeter has two different cells too, and one is out of calibration (or damaged somehow) while the other is not.