While in Madrid we will be doing a day trip to Toledo.

Depending on timing, we may have another day available to take a trip out of the city.

Some of the options I have looked at are Siguenza, Segovia (the closest) and Zaragoza.

For those that have been to any or all of these places, which would you recommend? If it helps, we will be there at the end of May/early June.

Segovia! I've been to all 3.

Tough call between Segovia and Zaragoza! But, Zaragoza probably deserves more time.

Zaragoza does deserve more touring hours to see the Basílica del Pilar, the cathedral, La Lonja and mudéjar architecture, but it's quite a longer day trip, as in one-way 1 hr. 15 to 1 hr. 30 min. via Ave, Avlo, the new Iryo or Ouigo, then touring all the sites.

Segovia is an easier 30 min. day trip from rail station Chamartín via Avant train. Much closer to Madrid.

It also depends on the time of year--Zaragoza can be super, super hot in the summer.
I find Zaragoza an easier trip from Barcelona, but I avoid it in the heat of the high summer.

Agree.
I visited Zaragoza for a day
( almost half way to Barcelona)
,wished I had more time .

Zaragoza for me would deserve an overnight----as danon says, hard to see it all in a half day.

Segovia !!