Sounds just like the 1st gear pop out.
have you changed your tranny oil?
it is easy, but a shop wouldn't charge much if you ha e a shop change your oil, or tires etc, just bring in your own oil.
I use royal purple syncromesh, I have 162k miles and have changed it 3 times, I look at the old oil for the shimmer and see how much metal is in it.

but with the 1st gear pop out, if you try and pay attention while you are shifting into 1st sometimes you can feel a double click going into 1st, when it pops out, is when it does not get completly engaged.
there are some tricks that help
example,
If at a stop, put in gear, take out and go back in, if you try to feel the shifter you can feel the double click on 1st.

when coming to a stop light or stop sign, as slowing down around 2 -4 mph if you slid it into gear, will go right in all the way.

There are seveal others.


but a good oil change and could help also, can fix alot of shifting problems in any styles of manual transmissions.

and do not let them put in 80-90, 75-140 etc which is normal gear oil and it will eat up the brass syncros, many larger manual transmissions take gear oil, and have heard several try to tell me that is what to use, but even if you read the bottles they say not to use with brass.