Are there ways to estimate size of the "whole universe"? Words escape me, but by "whole universe" (I think) I mean everything that's spatially connected to the observable universe in a conventional sense. If there is a better term for it, please let me know!

I was surprised to find out that theories and measurements about the big bang seem to mostly relate to the size of the observable universe via expansion, and if I understand correctly don't say much about the size of the whole universe.

Have I got this right? Are there any ways at all to try to estimate size of the whole universe? Does the term even mean anything?

See @RobJeffries' excellent answer to When will the number of stars be a maximum? and @Acccumulation's comment for background on this question.
tl; dr The universe is probably infinite, but if that's the case it's impossible to verify. If the universe is finite, and small enough, and the global curvature is equal to the curvature of our observable universe, then we will be able to estimate its size.

If the global curvature of the universe isn't positive, then the size of the universe is infinite, (and it's always been infinite since the dawn of time, at the Big Bang), assuming the topology of the universe is trivial.
Measurements of the observable universe indicate that the curvature may be zero (giving a flat universe), or even negative; if the curvature is positive, then its value is very small. We assume that the observable universe is typical of the whole universe, but of course that's impossible to verify.
Wikipedia says that

experimental data from various independent sources (WMAP, BOOMERanG,
and Planck for example) confirm that the observable universe is flat
with only a 0.4% margin of error.
[...]
The latest research shows that even the most powerful future
experiments (like SKA, Planck..) will not be able to distinguish
between flat, open and closed universe if the true value of
cosmological curvature parameter is smaller than $10^{-4}$. If the true
value of the cosmological curvature parameter is larger than $10^{-3}$ we
will be able to distinguish between these three models even now.

So even if the total universe does have positive curvature, its size is much larger than the observable universe, assuming that our observable universe isn't a patch that's abnormally flat.
According to How Big is the Entire Universe? by Ethan Siegel, if the curvature is positive, then the diameter of the total universe is at least 14 trillion lightyears, but that article is from 2012, more recent calculations may give a larger value.