When do I have to pay for the Mexican tourist card?

Does anyone have suggestions about how to avoid paying twice?

tl;dr I don't

There is an image of such a card on Baja Bound Insurance Services Inc. where its title is "Forma Migratoria Múltiple (FMM)". Details there include:

• Cost as of September 2015, $332 Pesos (appx U.S. $20*) per person; FREE if the trip is 7 days or less and you cross by land
• May be issued for up to 180 days

and:

According to the INM, an FMM, or Forma Migratoria Múltiple is an "admission document" issued to vacationing visitors of certain nationalities. Although the tourist card (FMM) is also popularly known as a tourist "visa," it is not officially a visa. U.S. and Canadian citizens driving to Mexico for vacation may obtain a tourist card (FMM) within minutes at a Mexican immigration office at the border with only a valid passport or passport card. The tourist card application is now available online as well. The tourist card (FMM) may be issued for up to 180 days and is issued per person including children.

More relevant to this question (same site different page):

Is it possible to enter and exit Baja multiple times with the same tourist card (FMM)?

Yes. The INM delegate for Baja California (Norte) announced in September 2015 that the FMM is now multiple entry for land travel in Baja California only.

Noting the only the inference is that the card is single-entry elsewhere in Mexico. Instituto Nacional de Migración has, in connection with the electronic version of the form for entry by land:

El solicitante se da por enterado que la FMM tiene una vigencia máxima de 180 días naturales y podrá utilizarla para una sola internación.

The last part of which is "for single admission".

Undated information from mexonline has:

The card, known as the Migratory Tourist Form (FMT) is valid for six months (180 days) with multiple entries. Make sure you ask for the full 180 days even is you plan to stay only a short time. You never know if you may return in those 180 days.

but judging by the cost mentioned there (190 peso) the site is not up to date, though at least indicates that the FMT was indeed multiple entry.

* Mexperience which has approximately US$22 for the permit also has:

If you lose your FMM while you’re in Mexico, you will need to visit one of the immigration offices situated in towns and cities across the country, or at the airport, and apply for a replacement before you can leave. This will involve some form-filling and filing, and a trip to a local bank to pay your permit replacement fee (about US$40) before you return to the immigration office to receive your FMM replacement.

The same site states:

If you are driving out or leaving the country by sea, you should voluntarily surrender your FMM to an immigration official before your departure. Failure to do so might cause delays the next time you try and enter Mexico.

There is even advice that if you leave the county with it by mistake you should contact a Mexican Consulate for what to do with it.