Hi Jim,
please have a look at the attached pic of a 350 EW gear cluster.
The best thing would be to strip the gearbox and check the dogs for condition. If you find badly rounded dogs at the spot indicated by red arrows on the left, this is the reason for your problem and there is probably not much to do than to have these dogs re-ground by a competent machinist.
The dogs of the 2nd gear wheel on the shaft, see arrow in the middle are also prone to wear. They need to be a bit rounded at their ends to facilitate selecting 2nd gear, but should be parallel in the middle.
First thing, as stated before, is to make sure that the dogs of the 3rd gear wheels are fully engaged. You can check that by taking one of the clevis pins out of the linkage while 3rd gear is engaged and look if you can move the part of the linkage that goes to the gearbox any further. If so, try to adjust.
If this cures 3rd gear but you find that now 2nd gear has a tendency to jump out it might be the fault of all the clevis joints having too much play. The obvious cure is to bore or ream them out and make oversize pins.
If this does not help, you will need to strip the gearbox to investigate further.
Cheers
Wilfried