Does anyone know of a mid-day train that goes from the CT area to Innsbruck? All I see is one that leaves lA Spezia at 10 something in the Am or the overnight train at 4:30pm. Any other ideas? Thanks!

I suspect you are looking at RailEurope...which only shows SOME trains.

Try the German rail site for complete listings:
http://www.bahn.com/i/view/GBR/en/index.shtml

When boarding an Italian train, I prefer to search the Trenitalia website. It gives you fares on the options that are all-Trenitalia while bahn.com does not.

http://www.fsitaliane.com/homepage_en.html

Use www.bahn.de/en to check overall journey options - personally, I'd search with 'duration of transfers standard' changed to 'at least 45 minute'.

Identify a suitable option, with the fewest connections .

Now split the booking and book CT to Verona at the Italian Railways site www.trenitalia.com from €19 and the Verona to Innsbruck train from €39 at www.oebb.at

La Spezia dp 12.26 or 14.22 - Innsbruck ar 22.08 resp 23.44, change at Parma, Bologna and Brennero
La Spezia dp 12.40 or 14.40 - Monterosso dp 12.55 or 14.55 - Innsbruck ar Innsbruck ar 22.08 resp 23.44, change at Milano, Verona and Brennero.
Overnight connection:
La Spezia dp 18.54 - Firenze 21.12/22.09 - Innsbruck ar 4.32