Switching companies during training?

Don't go with the Construction Company, unless that is what you want to do. The experience is not considered in the OTR world (even if you cross state lines)---don't ask me why.
And if you stay with the construction company for, say, a year...all you have (according to the majority of trucking companies) is a year old CDL, and a year since you've gotten it. There is a good chance you would have to start all over...or at the least, a refresher course and trainer.
Seen it happen in real life, and right here on this site as well.

 

Also, don't wait for that phone call. Chances are they have a lot of new grad applicants and sometimes they just want to see some motivation on your part besides filling out a 10 minute application.

I had a friend get me on at my gig when I started as a new cdl driver, had a lot of prehires but never heard back from the job I really wanted. Called them up for two days in a row and was off working for them and haven't looked back. It gave me two extra days to fish too, those were the days.