The big issue you're going to run into is the need for a hotspot. While there are still recycled and cheap phones made by one or two manufacturers which possess the other features, it's very rare for any of them to also serve as a hotspot. They are considered entry-level phones and usually use a proprietary operating system so everything "looks like the ...
I like "small" phones too, mine is a 4G / 4GB / 64GB 5' Redmi with Android. Couldn't find smaller that fits my needs. It's powerful, reliable, paid $120 for it, even if too big for you. But a quick search where I bought mine brought the following:
Melrose - 3.4" - Android 8 1
SOYES XS - 3" - 4G LTE - Android 6.0 - 2GB / 16GB
U2 - 4G Lte - 3.15" - 1GB / 8GB -...
The Huawei E5788u-96a is a LTE router with CAT16 speeds
Another option is the Huawei H112-372
Seems that Huawei, Netgear, and HTC are the only companies at the moment that make gigabit LTE routers.
Most smartphones brands are making phones bigger and even Samsung phones, they're all 5 inches or bigger.
With that said, you have a Sony Xperia XZ2 Compact which costs about 430 EUR. It's a compact phone with a 5-inch display.
You also have the Palm Phone which is a 3.3-inch screen; it got Android and it's functional for your needs but the Palm Phone has ...
Any mod capable Moto Z series device can be equipped with a Livermorium sliding keyboard mod, although as of answering this question it is out of stock from the manufacturer but should be available by years end again.
The big issue here is the requirement for a hotspot, which most older phones would not support properly... So stepping up to a newer device ...
You may want to consider the Microsoft Surface 3 LTE. I have one myself, and really like it. I specifically purchased it for it's built-in LTE capability, because I find looping WiFi through a phone or using a dongle awkward. Unfortunately, Surface 3 is no longer available direct from Microsoft - I don't know what the other options in your area are. You ...
I'd go with a standard home router with the following constraints:
supported by OpenWRT.
comes with at least one USB port.
then plug a USB-4G dongle into it. See http://wiki.openwrt.org/doc/recipes/3gdongle for some info about how to configure OpenWRT for that kind of use.
Which specific router to use will depend on what other features you want it to ...