Antonym for "thimbleful"

Solution 1:

If you really need a t-word, you could go for torrent, defined by the online Merriam-Webster as:

1 : a tumultuous outpouring : rush

2 : a violent stream of a liquid (such as water or lava)

3 : a channel of a mountain stream

Torrent is often used to describe large amounts. For example, buried under a torrent of words or One tweet releases a torrent of stories Since thimble is often (usually?) used to describe small quantities of liquid, the juxtaposition can work:

I had a thimble of patience and the task required a torrent!

On the other hand, torrent carries the implication of something arriving with force, suddenly, so it isn't perfect.

Another alternative, a bit more prosaic, is simply ton:

I had a thimble of patience and the task required a ton!

Or:

I had a thimble of patience and the task required a metric ton!

The word metric here, is used as an intensifier¹, so that makes it a slightly stronger statement. In a similar vein, if a bit more crude, you can go for crapton. This is a slang and relatively vulgar term which isn't in any serious dictionary I could find, but is defined by Wiktionary as:

(slang, vulgar) A very large amount.

So, you could use:

I had a thimble of patience and the task required a metric crapton!

¹_{I admit I have no evidence of metric as an intensifier other than my own experience, but I am reasonably confident it is used this way. I will admit that this might just be because of the recent popularity of metric crapton though.}

Solution 2:

A possibility is truckful. It starts with a T, and it has a similarity in its ending sound (though not fully rhyming).

informal a very large amount of something

Example: I had a thimble of patience and the task required a truckful.

The connection would be stronger if you used thimbleful instead of thimble.

A very small quantity.

Example: I had a thimbleful of patience and the task required a truckful.

The juxtaposition of these two words is not unknown, as shown by these usages:

Springfield News Leader

The probably 4-foot-11 Roder may possess a thimbleful of height, but she makes up for it with a dump truck full of heart

How to Tap Into the Goodness of God

God always multiplies, but wouldn't you rather have a truck full instead of a thimble

What may be more common though, is truckload, instead of truckful, as in these examples:

Zil Thrills in the '70s

It was a time when simple information was doled out by the thimble, and fantasy by the truckload.

South Jersey deals with higher levels of radioactivity in drinking water

It’s not like you’re going from a thimble full to a truckload full.

Solution 3:

"I had a thimble of patience and the task required a tun."

tun (plural tuns)

1. A large cask; an oblong vessel bulging in the middle, like a pipe or puncheon, and girt with hoops; a wine cask.
2. (brewing) A fermenting vat.
3. An old English measure of capacity for liquids, containing 252 wine gallons; equal to two pipes.

Obviously this word eventually became the more commonly seen "ton", but has a stronger connotation of the container itself as opposed to the volume or weight measurement, paralleling your use of "thimble".