Yes, if there's crud in the valve body or passageways they may be clogging or making things stick until the oil warms and thins out.

There may also be gunk inside the transmission pressure switch assembly - behaving the same as above but instead mis-reading the pressures inside the transmission until the temperature is up.

I found a writeup on disassembly of the transmission:
4L60E Disassembly

You can follow it down and click on the pics to get an idea of the area to go looking at - when it starts into "Remove the pump housing bolts" - you're into stuff that can only be done with the transmission out. You can hit 'next' onto the assembly guide and scroll down the valve body section for more pics of the area too

HERE's a pic of a valve body exploded out into its parts (bottom round thing is the pump - some of the parts in it are accessible when the valve body is off.)

At the end of each row of parts there is a clip shown: You don't have to pull everything out of the valve body at once - the clips may be removed one at at time, things cleaned, inspected, re-installed and then move onto the next 'row' of parts.

On the right hand side of the above pic there is a solenoid labelled 'B' and the row of parts beside are for the 2-3 shift - this is where I'd start looking for sticking/crud, I've been told if one is very careful and clever the solenoid and parts can be removed with the valve body still in place - but I don't know I've never tried.

The simplest easiest cheapest step would be to 'service' the transmission yourself - get a kit (around $20 up here in canukland) - comes with pan gasket and new filter - undo the bolts on the pan, pull down one corner (helps to leave the bolts in but loose along one side) and drain the oil into a large shallow tub. Pull the pan the rest of the way off (pour out the rest of the oil) and start looking for crud and gunk. Filter pulls straight down - leave the seal for the neck alone unless its damaged, apparently they're a bugger to get in and out. My pan had a small flat magnet epoxied to the bottom - it was covered in a layer almost 1/4" thick of - well - ferrous gunk, but then I had no 2nd 4th or reverse (sunshell teeth stripped out) - that's a thumbnail sketch of what's involved but I'd go find a proper writeup with all the details

You'd be looking for indications of the state of the transmission based on the amount/type of gunk found, also a chance to look at what you 'might' end up looking at in closer detail.

Even if there is some stuff in the oil, you may find that fresh(er) oil (only about half gets drained out - rest sits up top and in the torque converter) and a new filter may improve things - it could be restriction in the filter that's gumming up the works till it gets warm too.

 

I have a 2001 4wd blazer that won't shift out of 3rd when it's wet out. Any advice?