If your thermometer takes 60-90 seconds, it's definitely time to consider a new thermometer! There is a wide variation in times that oral/rectal thermometers take to read. You can get quick read oral/rectal digital thermometers that will read the temperature in as little as 8 seconds. Old fashioned mercury ones are quite slow comparatively.
The American Academy of Pediatrics official stance is that rectal temperature is the most accurate for children under three, so it is a good idea to have a quick read thermometer for only rectal use for those times when you are concerned about "is that fever high enough I should take my child to the emergency room". Quick read will be really helpful compared to your slow-read one when you're trying to pin them still enough to get a reading at all.
However, the rest of the time for screening purposes, before getting out the rectal thermometer, using a thermometer that is more pleasant/faster/etc may be preferable, and (because of cross-contaminaton) you should never use the same thermometer for both oral and rectal temperatures anyway.
For a screening thermometer, there are many choices in addition to a quick read digital thermometer. There are temporal artery thermometers that you swipe across the forehead that read the temperature in about 6 seconds, and ear thermometers that read the temperature in about 3 seconds, as well as slow but more accurate pacifier thermometers for babies that read in about 90 seconds.
Ear thermometers do have to positioned facing the right direction to work, and may give false highs if your child has an ear infection, but are so fast and easy you can take the temperature several times should you be concerned you might not have positioned it right, and are apparently accurate enough that even our usual urgent care facility prefers to take their screening temperatures using the ear thermometer. Used correctly they're usually consistent, but may vary per operator. We like our ear thermometer, but some models are also less good for babies under three months.
Temporal Artery/Forehead Thermometer - Non-invasive, can be done even while baby is sleeping, fairly quick. But they're generally expensive. And they're less likely to be accurate or consistent than an oral/rectal temperature. Slightly slower than an ear thermometer, but the low invasiveness level may make it more tolerable.
Armpit thermometer - Many oral/rectal thermometers can also take armpit temperatures. But it is less accurate again as a measuring technique. But for screening purposes, it is an option that can be considered.
So, in short, there are many choices, but the first priority to have on hand should be a quick-reading rectal thermometer, however, if you can afford multiple thermometers, you may want a second quick-reading thermometer that is a more pleasant to use, despite its lower accuracy. The price varies significantly between types and brands of thermometers, and price is not a guarantee of accuracy. Consumer reports has done tests to determine which specific brand(s) are most accurate/fastest/easiest/etc. however, one must be a subscriber to see the results.