Is there a straightforward word for "The thing in between first and second"?

The word you are looking for is the sesquialter, sesquialteral, sesquialterate, sesquialteral, or sesquialterous element.

You can also use sesquialter as an ordinal noun, for the midpoint between first and second:

I’ve skipped the zeroth, but then jumped straight from the first to second, so I’m am still missing the sesquialter I expected midway between those two.

That’s because sesquialter means “one-and-a-halfth”,¹ but is substantially easier to say.² It is one of those sesquipedalian terms surpassing both in erudition and utility alike. Per the OED, it means:

1. Of a proportion: That is as 1½ is to 1. Of an object: Proportionate to another object as 1½ is to 1; that is such a multiple of.

Here are a few of its more recent citations:

• 1698 Phil. Trans. XX. 81 ― We assign to a Fifth··the Sesquialter Proportion (or that of 3 to 2).
• 1711 H. Needler in J. Duncombe Lett. (1773) I. 90, ― 6 is only sesquialter of 4.
• 1715 Cheyne Philos. Princ. i. 222 ― In all the Revolutions of the Planets about the Sun,··the periodical Times is [sic] in a Sesquialter Proportion to the middle Distances.
• 1784 J. Keeble Harmonics 29 ― The sesquialter chromatic.
• 1846 Penny Cycl. Suppl. II. 369/2 ― The following ratios are super‐particular: 15 to 10, which is sesquialter.

And its etymology is:

Etymology: L., f. sesqui- (see prec.) + alter second. For the formation cf. ONor. hálfr annarr, OE. óþer healf, G. anderthalb.

The sequi- prefix is today most familiar in terms like sesquicentennial, for the 150-year anniversary of some event. It gives rise to a delightful multitude of derived terms.

1. a. With designations of measure or amount, denoting one-and-a-half times the unit; as sesquihōra an hour and a half; sesquipēs a foot and a half (see sesquipedalian); so † sesquiˈhoral a., lasting an hour and a half; ˌsesquioˈcellus Ent. (see sesquialterous); † ˈsesquitone Mus., an interval consisting of a tone and a semitone, a minor third; also used loosely in † sesquiˈdecuman a., consisting of fifteen; † sesquiˈdecury, a set of fifteen.

The OED provides not only senses 1b through 1d, with which we need not here concern ourselves, but also the operative sense 2a:

2. a. With an ordinal numeral adjective, denoting the proportion 1 + ¹⁄_n:1, i.e. n + 1:n, where n is the corresponding cardinal number, as sesquioctāvus, bearing the ratio 1⅛:1, i.e. 9:8; so sesquialter, -altera, etc., sesquitertia, etc.; † ˌsesquibiˈtertial, involving a proportion of 5:3; † ˌsesquiˈdecimal, of 11:10; sesquiˈnonal, of 10:9; ˌsesquiocˈtaval, -ˈoctave, of 9:8; ˌsesquiˈquartal, -quartan, of 5:4; ˌsesquiˈseptimal, of 8:7.

Sense 2b also gives several nice words for harmonic situations:

b. in Music, after sesquialtera and sesquitertia; sesquiquarta, -quinta, -sexta, -octava (-octave), -nona, applied

• (i) to harmonic intervals producible by sounding four-fifths, five-sixths, etc. of a given string;
• (ii) rhythmic combinations of four notes against five, five against six, etc.

As you correctly perceive, having a word that means ³⁄₂^th is exceedingly convenient at times, which no doubt why are ancestors invented sesquialter way back in the 16^th century. This is the first citation given for the term in the OED:

• 1570 Dee Math. Pref. c j b, ― A Cylinder, whose heith, and Diameter of his base, is æquall to the Diameter of the Sphære, is Sesquialter to the same Sphære.

Footnotes:

1. Regarding the productivity of the -th suffix for creating ordinals from cardinals in English, see this question.

2. Insofar as words that have end in -fth or -xth are often considered difficult to pronounce by non-native speakers, like fifth, sixth, twelfth, and if they find twelfth tough, they seem likely to find halfth tough as well.

If you google halfth, you get 114,000 hits, the first of which is "One-and-a-halfth-order logic." And there's the famous cartoon Duck Dodgers in the 24½th Century (the official pronunciation of this is twenty-fourth and a half).

UPDATE:

Googling, I get two hits for "one-and-a halfth derivative" and three for "first-and-a-half derivative". So these are both possibilities.

This is a hard one, and maybe you would have better luck at mathoverflow (but do link to here, so that they don't send you away).

In mathematical papers it is not uncommon to find constructs which bring out situations that common language would not encounter (or that would be so rare that short of a poem you would have trouble find it).

Back to your issue - it is really not uncommon to find and see as completely normal linguistic monstrosities such as for example

... of the (α+1)-th order ...

which seem natural when reading the paper, but if you stop to consider how you would pronounce it if you were presenting the paper you are soon to realize that you would probably rephrase it on the spot to something else.

So, strictly mathematically speaking, if the above notation is acceptable to you then, mathematically, since α+1 can be anything then for α = 1/2 it is equal to 3/2, so for that case you would write

... of the (3/2)-th order ...

But now the problem of pronunciation is even more proclaimed.

I did find an article called one-and-a-halfth-order logic in oxford journals.

Though I did not find dictionary definitions of halfth I did find quite a few results for it in books.